Shock and Vibration 10 (2003) 269–279 269IOS Press

Dynamic analysis of shell structures withapplication to blast resistant doors

C.G. Koh∗, K.K. Ang and P.F. ChanDepartment of Civil Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260

Received 2 July 2001

Revised 24 September 2002

Accepted 2 May 2003

Abstract. This paper concerns the dynamic analysis of shell structures, with emphasis on application to steel and steel-concretecomposite blast resistant doors. In view of the short duration and impulsive nature of the blast loading, an explicit integrationmethod is adopted. This approach avoids time-consuming computations of structural stiffness matrix and solving of simultaneousnonlinear equations. Single-point quadrature shell elements are used, with numerical control to suppress spurious hourglassmodes. Composite shells are handled by an appropriate integration rule across the thickness. Both material and geometricnonlinearities are accounted for in the formulation. Contact and gap problems are considered using bilinear spring elementsin the finite element analysis. Numerical examples are presented for some benchmark problems and application study to blastresistant doors. Good correlation is generally obtained between the numerical results based on the software developed and theresults obtained by other means including field blast tests.

Keywords: Dynamics, vibration, shell, blast doors, composite, steel, concrete

1. Introduction

Blast resistant doors are commonly used in defenceshelters, ammunition storage dumps and blast-resistantstructures alike. These doors have to be designed towithstand blast loading and shock waves, so as to pro-tect occupants and objects inside, and in the case ofammunition storage, to prevent chain reaction of ex-plosions from one compartment to another. Inadequatedesign of the blast resistant door would undermine theoperational performance and survivability of the struc-ture and users. On the other hand, an overly conser-vative design would result in incurring unnecessarilyhigh costs.

Recent studies on blast resistant structures mainlydealt with reinforced concrete structures, see [1–5] forexample. Lok and Xiao [6] extended the study to steel-

∗Corresponding author. Tel.: +65 68742163; Fax: +65 67791635;E-mail: cgkoh@nus.edu.sg.

fibre-reinforcedconcrete panels subjected to blast load-ing. Krauthammer carried out numerical studies onthe blast response of structural concrete and structuralsteel connections [7]. The present study focuses on thedynamic analysis of steel blast doors and compositeblast doors in the form of steel-concrete-steel sandwichpanels.

Geometrically, blast resistant doors may be flat orcurved, though the doors are usually rectangular infront elevation. The structural resistance of blast doorsis primarily provided by both membrane and bendingactions. Correct modelling of the nonlinear behaviourof blast resistant doors under severe loading is there-fore essential in their safe and cost-effective design.Dynamic analysis of shell structures, among all struc-tural forms, is perhaps the most challenging both math-ematically and physically. Mathematically, it is rel-atively difficult to formulate and obtain solutions forshell structures, particularly when nonlinear dynamicresponse is involved. Physically, the response of shellstructures involves the interaction between the mem-

ISSN 1070-9622/03/$8.00 2003 – IOS Press. All rights reserved

270 C.G. Koh et al. / Dynamic analysis of shell structures with application to blast resistant doors

brane mode and bending mode which, in general, areof different orders of magnitude in stiffness effects.

Analytical approaches do not apply to nonlinearproblems in general, and numerical approaches have tobe used. With rapid advances in computer hardware,the finite element method has proven to be a versa-tile and powerful tool for nonlinear dynamic analysisof shell structures. Many shell theories and solutionmethods have been developed in the last few decades.The finite element formulation of shell elements canlargely be categorised into two approaches: (a) directapproach based on a selected shell theory and (b) ap-proach based on degenerated continuum element firstproposed by Ahmad et al. [8]. A good literature reviewof shell elements may be found in [9].

As this study involves short-duration loading, ex-plicit dynamic analysis, which avoids the time consum-ing simultaneous equation solver, is preferred. For ex-plicit analysis, 4-node shell element derived by the de-generated continuum approach is an attractive choice.In contrast, higher-order elements are sensitive to meshdistortion and less stable in an explicit dynamic anal-ysis [10]. Reduced integration with single quadra-ture point is preferred for reason of computational ef-ficiency, but numerical means have to be introduced tosuppress hourglass modes.

In blast resistant door designs, steel and concrete ma-terials are commonly used. A combination of steel andconcrete would take advantage of high strength of steeland added inertia of concrete in-fill. The sandwich-likestructure would require composite elements in the anal-ysis. In this regard, the explicit dynamic approach de-veloped by Koh et al. [11] for 4-node composite shellsis adopted herein for the dynamic analysis of blast re-sistant doors. The constitutive models adopted for steeland concrete are, respectively, Von-Mises yield crite-rion with hardening and a scalar-damage model (es-tablished in the framework of continuum mechanics).Bilinear spring elements are used to handle contact andgap between the door and door frame. Nonlinear effectsdue to geometry updates are accounted for by means ofthe updated Lagrangian formulation.

2. Composite shell elements and dynamic analysis

In this study, a bilinear four-node quadrilateral shellelement with single quadrature point [12] is adoptedin the finite element modelling of blast resistant doors.In essence, a co-rotational coordinate system is estab-lished for each element. The nodal coordinates and ve-

locities are transformed to the co-rotational coordinatesystem. The velocity strains for membrane stretch-ing, bending and shear are computed. This involvesstraightforward strain-displacement matrices evaluatedat the origin of the co-rotational coordinate system. Asthe element is under-integrated (in plane), a stabiliza-tion approach called perturbation hourglass control asproposed in [12] is adopted to suppress the spurioussingular modes.

Since blast resistant doors may be composite in ma-terial, it is necessary to split the element in severalplies that are not necessarily made of the same ma-terial. Different plies may have different thicknesses.In this regard, the approach for the explicit dynamicanalysis of laminated composite shells [11] is appro-priate. It is assumed herein that the bonding betweenplies remains intact with no delamination even thoughdeformation may be large. The focus of the analysisis the global behaviour (e.g. deflection) of the struc-ture, rather than the local behaviour (e.g. strain vari-ation through the thickness). Therefore, the assump-tion that the plane section across the thickness remainsplane is made. To account for possible large thicknessstrain, the thickness is updated using log-strain ratherthan engineering strain. Geometric nonlinear effectsdue to large deformations are included by means of theupdated Lagrangian formulation.

In terms of dynamic analysis, there are two possi-ble approaches: the implicit and explicit integrationmethods. In the implicit methods, element stiffnessmatrices have to be computed, assembled and stored inthe global stiffness matrix. The equations of motionare converted into a set of algebraic equations by somenumerical integration schemes such as the Newmark-Wilson family [13] and then solved by a simultaneousequation solver. Since the system is nonlinear, an iter-ative scheme such as the Newton-Raphson method isrequired in the equation solving until certain conver-gence criterion is met. As the system is nonlinear andcoupled, the equation solving process normally formsthe bulk of the computational effort. Thus, though theimplicit methods with appropriate parameters can beunconditionally stable, the cost per time step is oftenhigh.

Explicit methods, on the other hand, do not requirethe formation of stiffness matrices. The equations ofmotion in the global co-ordinate system may be writtenas

MU + CU(t) + P (U , t) = F (t) (1)

whereU , P , F , M andC are the displacement vec-tor, internal force and external force vectors, mass and

C.G. Koh et al. / Dynamic analysis of shell structures with application to blast resistant doors 271

damping matrices, respectively. Note that no stiffnessmatrix is required. With the assumption of lumpedmass matrix and mass-proportional damping matrix,the equations of motion are uncoupled. Expressing theacceleration and velocity in terms of displacement, thedisplacement vector can be solved for the current stepwithout a simultaneous equation solver. The drawbackis that explicit methods are conditionally stable; thusa small time step (smaller than a critical value) hasto be used. This, however, is not a hindrance in thisstudy because the time step used for impulsive (blast)loading is normally small in order to accurately capturethe loading function. In such applications, the over-all computational cost for explicit methods is generallyless than implicit methods.

In the present study, a simple yet effective explicitmethod known as the central difference method is cho-sen [13]. Let subscriptsn andn− 1 denote the currentand previous time steps, respectively. The velocity forthei-th degree of freedom is replaced by the followingcentral difference approximation:

Ui,n+0.5 =Ui,n+1 − Ui,n

∆tn(2)

where ∆tn denotes time step used at then-th step(current step) and subscriptn+0.5 denotes the specifictime midway between then-th step and(n+1)-th step.Similarly,

Ui,n−0.5 =Ui,n − Ui,n−1

∆tn−1(3)

Note that the formulation allows variable time steps.For certain situations where long-term response isneeded, the computational time would be reduced sub-stantially with the application of a variable time stepscheme. The velocity and acceleration at the currentstep are given by:

Ui,n =12(Ui,n+0.5 + Ui,n−0.5) (4)

Ui,n =1

∆tn(Ui,n+0.5 − Ui,n−0.5) (5)

where

∆tn =12(∆tn−1 + ∆tn) (6)

Making use of the above equations, it can be readilyshown that Eq. (1) yields

Ui,n+0.5 =(

2Mi − Ci∆tn2Mi + Ci∆tn

)Ui,n−0.5

(7)

+(

2∆tn2Mi + Ci∆tn

)(Fi − Pi)

ε

σ

To account for gap

Fig. 1. Stress-strain relationship of contact/gap element.

As the explicit scheme is conditionally stable, the timestep has to be less than a critical value to ensure numer-ical stability. The critical time step is computed basedon a safe estimate as follows [14]

∆tcritical = (8)

Ls

[ρ(1 − v2)/E

2 + 0.83(1 + 1.5L2s/h2)(1 − v)

]0.5

whereρ is the density,E Young’s modulus,v Possion’sratio,h element thickness, andLs the shortest distancebetween adjacent nodes.

The stresses are computed at all the integration pointsthrough the thickness. In general, composite layerthicknesses are arbitrary and need not necessarily besymmetrical with respect to the mid-surface of the ele-ment. Each layer can be further divided into sub-layers(of the same material), depending on the relative thick-ness and desired numerical accuracy. In this instance,the commonly used rule of Gauss quadrature is notconvenient for integration through the thickness. Fur-thermore, this rule is not necessarily the most accuratestrategy for integration of nonlinear stress through thethickness [15] and it is thus justifiable to use a simplerintegration rule. In the present study, the mid-pointrule is used, i.e. integration points are the midpoints ofsub-layers and quadrature weights are proportional tothe sub-layer thicknesses, though not equal in general.For ease of reference, this is termed as the compositeintegration rule. This is different from the centroidalmethod as presented in [15], which divides the elementthickness into sub-layers of equal thickness.

3. Material models

In the design of blast resistant doors, steel and con-crete materials are commonly used for reason of cost-effectiveness though more advanced materials are avail-

272 C.G. Koh et al. / Dynamic analysis of shell structures with application to blast resistant doors

Dynamic analysis

Explicit approach

Finite elementmodelling

4-node shellelement with

single quadrature &hourglass control Concrete:

Scalardamagemodel

Materialmodels

Steel:Von-Mises

model

Contact/gapelement

UpdatedLagrangian

Software BASS

Compositelayers

Fig. 2. Main features of software development.

able. The respective constitutive models are Von-Misesmodel and a scalar damage model. The hybrid use ofsteel and concrete would take advantage of the highstrength of steel and large inertia of concrete in-fill toenhance the performance of blast resistant doors. Thefollowing two types of blast resistant doors are consid-ered in this study.

(A) Steel-air-steel (SAS) door – The door comprisestwo “skin” plates, stiffened by side plates alongthe four edges and some stiffener beams (e.g.I or C-channel cross section) at intermediatelocations. All the plates and beams are made ofsteel.

(B) Steel-concrete-steel (SCS) door – The door issimilar to the SAS door, except that the void isfilled with concrete.

3.1. Elastoplastic model with hardening for steel

Steel usually exhibits a work-hardening behaviourwhen stressed into the elastoplastic zone, and theisotropic Von Mises material model is appropriate. Theyield function is generally in the form offy(σ, κ, . . .),whereκ is the hardening parameter which depends onthe accumulated plastic strain in some manner. TheVon Mises criterion for the assumed plane stress statecan be written as

F (σ, κ) = f(σ) − σ2Y (κ) (9)

whereσY is the yield stress in pure shear, and the yieldfunction(f) is given by

f(σ) = σ2x + σ2

y − σxσy + 3σ2xy (10)

For stress update, the key issue is often the integra-tion of the flow rule in each time step. A commonlyadopted approach is the backward Euler method whichis stable and usually efficient [16]. This involves itera-tive solutions to achieve convergence by means of, say,the Newton-Raphson scheme which is adopted in thisstudy. Non-iterative schemes are also available if theadvantage of high concurrency for the explicit dynamicanalysis is to be exploited towards vectorization andparallel computing [11].

3.2. Scalar damage model for concrete

The concrete model adopted here is a scalar dam-age model developed in the framework of the contin-uum damage mechanics [17]. It is relatively simplein formulation and easy to implement in the finite ele-ment analysis. More importantly, the strain rate effecton concrete behaviour is accounted for. The compu-tational model is partially verified by an experimentalinvestigation program involving impact tests of con-crete specimens. This model is appropriate for steel-concrete composite doors where concrete in-fill servesmainly to increase the inertia (rather than strength) andsevere concrete damage is not expected.

In general, the mechanical properties of concrete areenhanced with increasing strain rate. The domain ofstress and strain are divided into two parts, one for thelow (quasi-static) to intermediate rate(ε � 30s−1, andthe other, from intermediate to high rates. The CEB-1988 recommendations [18] are adopted with regards to

C.G. Koh et al. / Dynamic analysis of shell structures with application to blast resistant doors 273

start

input data

assemble mass matrix

compute mass-proportional damping matrix

compute external loads

compute internal forces material models

compute hourglass forces

update effective nodal forces

compute hourglass forces

output results

save data for future analysis

end

t = t + ∆t

compute vel/disp/acc.

Fig. 3. Computational procedure of dynamic analysis.

the rate-dependent concrete material properties undercompressive and tensile actions.

The rate dependent damage model is formulated us-ing the concept of internal variable to represent the ex-tent of concrete damage. The stress-strain relationshipof the concrete model is given as

σij = (1 − D)Eijklεkl (11)

whereσij and εij are the components of the stressand strain tensors, respectively,Eijkl denote the initialmoduli, andD is a scalar damage parameter that rangesfrom 0 (virgin material) to 1 (asymptotic failure). In

this concrete damage model, concrete is assumed toremain isotropic up to failure. The positive strains de-termine the growth of damage that is mainly associatedwith opening of micro-cracks. The damage consists ofboth tension and compression components, defined asfollows

D = αtDt + αcDc (12)

where subscriptst andc denote tension and compres-sion, respectively, andαt andαc are non-dimensionalfunctions of the principal strains. More details on thedamage model adopted can be found in [17] whichcovers experimental verification of the model.

274 C.G. Koh et al. / Dynamic analysis of shell structures with application to blast resistant doors

(a) Geometry of the spherical cap (b) Finite element mesh

Symmetry

SymmetryP

2θ

Fig. 4. Geometry of the spherical shell and its finite element mesh.

-0.12

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

0 0.2 0.4 0.6 0.8 1

Time ( ms )

Dis

pla

cem

ent

( in

)

Liu (1985)present solution

Fig. 5. Vertical displacement at centre of spherical cap – linearly elastic material.

3.3. Contact/Gap modelling

There are two types of blast resistant doors in prac-tice, namely the air-tight door and blast-tight door. Inthe former case, rubber gaskets are used between thedoor and door frames for airtightness, giving rise to astructural gap between the door and the door frame.In the latter case, the door is in contact with the doorframe (no gap) when closed. In either case, the con-tact and gap behaviour is modelled by using nonlin-ear spring elements in the finite element analysis. Thestress-strain relationship of a typical contact/gap springis illustrated in Fig. 1. The tension part represents thesituation when the door and the frame are not in con-tact, causing no stress in the element. When the doorcomes into contact with the door frame, the elementbecomes very stiff under compressive action.

4. Numerical study and discussion

Instead of using a general purpose software, a spe-cific PC-based software is developed as a research toolto implement the explicit dynamic analysis of blast re-sistant doors with the main features as illustrated inFig. 2. In particular, the composite shell element asmentioned earlier is incorporated. The flowchart of thesoftware called BASS (Blast Analysis of Shell Struc-tures) is illustrated in Fig. 3. The software has the op-tion of using Gauss integration rule for non-compositeshells or the composite integration rule otherwise. Thefollowing numerical examples are presented to verifythe formulation and implementation (more numericalexamples can be found in [19]).

C.G. Koh et al. / Dynamic analysis of shell structures with application to blast resistant doors 275

-0.16

-0.12

-0.08

-0.04

0.00

0.04

0 0.2 0.4 0.6 0.8 1

Time (ms)

Dis

pla

cem

ent

(in

)

Liu (1985)present solution

Fig. 6. Vertical displacement at centre of spherical cap – Von Mises material.

(a) (b)

Fig. 7. Single-panel blast door: (a) schematic diagram; (b) finite element mesh.

Example 1: Cantilever beamConsider a cantilever beam with rectangular cross

section and parameters as follows [12]: length=0.254 m, width= thickness= 2.54 cm, Young’s modu-lus= 82.8 MPa, density= 12.87 kg/m3 and Possion’sratio = 0.2. The beam is subjected to a suddenly ap-plied pressure loading is analysed using the softwareBASS. The applied pressure loading is 69 Pa (0.01psi), and the material is assumed to be linearly elastic.

The present solution agrees well with the published re-sults [12] as presented in Table 1. It is noted that thepresent solution is closer to the beam element solutionas compared to other solutions as summarised in [12].

Example 2: Spherical capClamped all around the edge, a spherical cap is sub-

jected to a suddenly applied pressure of 4.14 MPa (600psi). The geometric and material properties are as fol-

276 C.G. Koh et al. / Dynamic analysis of shell structures with application to blast resistant doors

Table 1Dynamic response of cantilever beam

Element type Number of elements Max. deflection (mm) Fundamental period (ms)

Euler beam element 5 0.646 5.812Triangular plate elements 20 0.612 5.662Quadrilateral plate elements 5 0.623 5.68Present solution (with software BASS) 5 0.65 5.835

Table 2Comparison of maximum displacements

Point Maximum displacement Maximum Displacementon Outer Skin Plate (mm) on Inner Skin Plate (mm)BASS LUSAS BASS LUSAS

A 12.9 8.2 12.8 8B 23.6 24.8 22.4 17.6C 12.1 6.9 12.1 6.1D 22.5 24.6 6.4 4E 1.1 1.1 1 1

lows [20]: internal radius= 0.5657 m, thickness=1.041 cm, semi-angle (θ) = 26.27◦, Young’s modulus= 72.4 GPa, density= 2778 kg/m3, Poisson’s ratio=0.3, yield stress= 169 MPa, and hardening parameter= 0.

By virtue of symmetry, only a quarter of the sphere isconsidered and is discretised into12 elements (Fig. 4).Both linear and nonlinear analyses are carried out. Theproblem was analysed by Liu [20] using 8-node de-generated thick-shell elements and an implicit dynamicanalysis approach. The same material model is adoptedby Liu [20] and in the present study, and no rate effectis considered in this example. Figure 5 shows the ver-tical displacement at the centre of the sphere for lin-early elastic material, in comparison with Liu’s result.For the nonlinear case, the Von Mises yield criterion asdescribed earlier is considered but without hardening.The corresponding dynamic response is presented inFig. 6 and, due to softening effect (yielding), is gener-ally larger than the response for linearly elastic mate-rial as shown in Fig. 5. Note that in [20], the resultswere computed using 6 layers (accurate up to 5th orderpolynomial in the thickness direction), whereas in thepresent study, 5 Gauss points (accurate up to 9th orderpolynomial) are used. Therefore, the present solutionshould be more accurate than the results given in [20].

Example 3: Numerical study of blast resistant doorIn this example, a single-panel rectangular SAS door

subjected to blast loading is analyzed. The door consid-ered is made of two 9-mm mild steel skin plates, stiff-ened by four side plates along the edges and three C-channels at intermediate locations, as shown schemati-cally in Fig. 7(a). The outer skin plate (in direct contact

0

200

400

600

800

1000

0 2 4 6 8 10

Time (ms)

Pre

ssu

re (

kPa)

Fig. 8. Idealized blast pressure for Example 3.

Length L = 2.2 mWidth b = 1.3 mYoung’s modulus E = 200 GPaDensity ρ = 7800 kg/m3

Poisson’s ratio v = 0.2Yielding stress σY = 374#MPaUltimate stress σult = 440 MPaHardening parameter κ = 1.65 × 10−4

with the blast pressure) and inner skin plate are mod-elled by the shell elements. Stiffeners are modelled asbeam elements in this study as they deform predomi-nantly in flexural bending, though shell elements mayalso be used. The door is surrounded by a rigid doorframe around all four edges, with two hinges on the leftedge. There is an initial gap of 1 mm between door anddoor frame. There is no initial gap between latch anddoor frame. Figure 7(b) shows the finite element meshused in the numerical study. The parameters used inthis example are listed below:

C.G. Koh et al. / Dynamic analysis of shell structures with application to blast resistant doors 277

-0.005

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0 2 4 6 8

Time (ms)

Dis

pla

cem

ent

(m) B

E

CA

D

Fig. 9. Displacements at selected points on the outer skin plate.

-0.010

-0.005

0.000

0.005

0.010

0.015

0.020

0.025

0 2 4 6 8

Time (ms)

Dis

pla

cem

ent

(m)

B

E

A

C

D

Fig. 10. Displacements at selected points on the inner skin plate.

An idealized pressure loading as shown in Fig. 8 isused. The displacements at critical points on the outerand inner skin plates are presented in Figs 9 and 10,respectively. The displacements at points A, C and Eat the stiffeners are smaller than the other two points(B and D) away from the stiffeners, as expected. It isalso observed that the outer skin plate undergoes largerdeformation than the inner skin plate.

Numerical results for up to 8 ms are obtained bythe software BASS. The maximum displacements ofthe skin plates are presented in Table 2. These resultsare found to be generally in good agreement with theresults obtained by an implicit finite element analysissoftware called LUSAS [21], particularly for the twomost critical points B and D where the deformations arelarge. In terms of computational time, BASS based on

0

200

400

600

800

1000

0 2 4 6

Time (ms)

Pre

ssu

re (

kPa)

Fig. 11. Idealized blast pressure used for Example 4.

the explicit approach is found to be about twenty timesmore efficient than the implicit approach to achieveroughly the same accuracy.

278 C.G. Koh et al. / Dynamic analysis of shell structures with application to blast resistant doors

Table 3Details of blast resistant doors used in field tests

Door specimens Plate thickness Clear distance between outer Stiffeners(mm) and inner skin plates (mm) (C-channels)

SAS1 and SCS1 6 50 50× 25× 3SAS2 and SCS2 6 100 100× 50× 5SAS3 and SCS3 9 100 100× 50× 5SAS4 and SCS4 9 150 150× 50× 5

Table 4Maximum and permanent displacements at centre of blast door

Door specimen Maximum displacement (mm) Permanent displacement (mm)Field test BASS Field test BASS

SAS1 >85 180 N.A. 150SCS1 42 36 N.A. 18SAS2 >85 71 N.A. 51SCS2 35 11 26 2SAS3 N.A. 45 33 28SCS3 10 6 1 1SAS4 38 24 N.A. 14SCS4 15 4 0 0

Note: N.A.= Not available due to instrumentation problem.

Example 4: Comparison with field test results forblast resistant doors

Field tests of blast resistant doors of both SAS andSCS types were carried out, each under 100 kg barecharge of TNT with a stand-off distance of 5 m. Thepeak pressure loading is computed by a computer pro-gram called CONWEP [22] and its shape is idealizedas linear, as shown in Fig. 11. The dimensions of alltest specimens are 2.2 m× 1.2 m, as shown in Fig. 12.Each door specimen comprises two skin plates, threeC-channel stiffeners, and side plates all around the fouredges. A total of eight specimens were blast testedin the field: four SAS doors and four SCS doors. Inthe case of SCS doors, plain concrete of grade 30 (de-signed to have compressive strength of 30 MPa at 28days conforming to British Standard BS 5328 [23]) wasused as in-fill. Table 3 shows the geometrical dimen-sions of the specimens. The specimens were verticallyheld by clamping their two shorter edges on a speciallyconstructed concrete supporting frame. Displacementgauges were installed behind the door specimen to mea-sure the maximum and permanent deflection of thespecimen. The field test results are presented in Table 4and compared with the numerical results obtained byBASS.

Response measurement in the field is generally a dif-ficult task and particularly so for blast tests where thepeak response occurs in a very short time. Some fieldtest results may be inaccurate due to measurement er-rors or even unavailable. In fact, some results for per-manent displacements were not available due to dam-

1200 mm

4 x 550 mm

C-channel stiffeners Side plates

Skin plates(top & bottom)

Fig. 12. Schematic diagram of blast door specimens for field tests(plan view).

age of instruments during the blast test. Furthermore,it should be noted that the specified blast time history(peak value generated by CONWEP and idealized lin-early) used in the numerical simulation studies differsfrom the real blast in the field tests. In this light, thedynamic analysis by means of software BASS givesreasonably good correlation with the blast test results,as shown in Table 4. The other possible source ofdiscrepancy is the modeling of actual boundary con-ditions, which play an important role in the displace-ment response of blast resistant doors. Both the nu-merical and field test results confirm the effectivenessof the concrete in-fill in that the displacement responseof SCS blast doors is considerably smaller than that ofSAS blast doors.

C.G. Koh et al. / Dynamic analysis of shell structures with application to blast resistant doors 279

5. Conclusions

This paper illustrates the use of explicit dynamicanalysis for the analysis and design of actual blast resis-tant doors. The main characteristic of the quadrilateralshell element used is that only single quadrature pointis needed in the plane of each element. This results in asimplified formulation and low computational cost perelement-time step. The shell element allows the exten-sion to treat composite materials using the compositeintegration rule.

In terms of nonlinearity, three aspects have been in-cluded. (A) For material nonlinearity, Von Mises yieldcriterion with isotropic hardening is adopted for steel,whereas a scalar damage model developed for concreteis used. (B) Geometric nonlinearity due to possiblelarge displacement motion is accounted for by means ofLagrangian formulation. (C) The contact/gap behavioris modeled using bilinear spring elements.

The computational procedure has been coded lead-ing to the software called BASS. The numerical studypresented includes comparison with published solu-tions for some benchmark problems of beams andshells, with another finite element (implicit) softwareand with field measurements for actual blast resistantdoors. In general, the reasonably good correlation vali-dates the numerical procedure and software implemen-tation. The software developed is useful in the para-metric studies to optimize the design of blast resistantdoors.

Acknowledgment

The research grant and technical assistance (particu-larly on design data and field tests) provided by the Min-istry of Defence, Singapore, are gratefully acknowl-edged.

References

[1] S.C. Woodson and S.A. Kiger, Stirrup requirements for blast-resistant slabs,Journal of Structural Engineering, ASCE 114(1988), 2057–2069.

[2] F.B.A. Beshara and K.S. Virdi, Time integration procedures forfinite element analysis of blast-resistant reinforced concretestructures,Computers and Structures 40 (1991), 1105–1123.

[3] P.T. Nash, C.V.G. Vallabhan and T.C. Knight, Spall damage toconcrete walls from close-in cased and uncased explosions inair, ACI Structural Journal 92 (1995), 680–688.

[4] R.K. Otani and T. Krauthammer, Assessment of reinforcementdetails for blast containment structures,ACI Structural Journal94 (1997), 124–132.

[5] T. Krauthammer and R.K. Otani, Mesh, gravity and load ef-fects on finite element simulations of blast loaded reinforcedconcrete structures,Computers and Structures 63 (1997),1113–1120.

[6] T.S. Lok and J.R. Xiao, Steel-fibre-reinforced concrete panelsexposed to air blast loading,Proceedings of the Institution ofCivil Engineers – Structures and Buildings 134 (1999), 319–331.

[7] T. Krauthammer, Blast-resistant structural concrete and steelconnections,International Journal of Impact Engineering 22(1999), 887–910.

[8] S. Ahmad, B.M. Irons and O.C. Zienkiewicz, Analysis of thickand thin shell structures by curved finite elements,Int. J. Num.Meth. Engrg. 2 (1970), 419–451.

[9] H. Stolarski, T. Belytschko and S.-H. Lee, A review of shellfinite and corotational theories,Computational Mechanics Ad-vances 2 (1995), 125–212.

[10] Y.G. Zhu and T. Zacharia, A new one-point quadrature, quadri-lateral shell element with drilling degrees of freedom,Com-puter Methods in Applied Mechanics and Engineering 136(1996), 165–203.

[11] C.G. Koh, D.R.J. Owen and D. Peric, Explicit dynamic analy-sis of elasto-plastic laminated composite shells: implementa-tion of non-iterative stress update schemes for the HOFFMANyield criterion,Comp. Mech. 16 (1995), 307–314.

[12] T. Belytschko, J.L. Lin and C.S. Tsay, Explicit algorithms forthe nonlinear dynamics of shells,Comp. Meth. Appl. Mech.Engrg. 42 (1984), 225–251.

[13] K.-J. Bathe, Finite Element Procedures, Prentice-Hall, NewJersey, USA, 1996.

[14] T.Y. Tsui and P. Tong, Stability of transient solution of mod-erately thick plate by finite difference method,AIAA Journal9 (1971), 2062–2063.

[15] C.J. Burgoyne and M.A. Crisfield, Numerical integration strat-egy for plates and shells,Int. J. Num. Meth. Engrg. 29 (1990),105–121.

[16] J.C. Simo and R.L. Taylor, A return mapping algorithm forplane stress elastoplasticity,Int. J. Num. Meth. Engng. 31(1986), 1069–1086.

[17] C.G. Koh, Z.J. Liu and S.T. Quek, Numerical and experi-mental studies of concrete damage under impact,Magazine ofConcrete Research 53 (2001), 417–427.

[18] Comite Euro-International du Beton (CEB), Concrete struc-tures under impact and impulsive loading,Synthesis Report.Bulletin d’Information No. 187, Lausanne, 1998.

[19] P.F. Chan,Dynamic Analysis of Shell Structures with Appli-cation to Blast Doors. M.Eng. Thesis, National University ofSingapore, Singapore, 1999.

[20] G.Q. Liu,Nonlinear and Transient Finite Element Analysis ofGeneral Reinforced Concrete Plates and Shells, Ph.D. Thesis,University College of Swansea, UK, 1985.

[21] LUSAS Finite Element Software, Version 12, FEA Ltd. Sur-rey, UK, 1998.

[22] Fundamentals of Protective Design for ConventionalWeapons, (Software CONWEP), US Dept of the Army, Tech-nical Manual, TM5-855-1, 1987.

[23] BS 5328: 1999 Concrete. Part 1: 1997 Guide to specifyingconcrete (amended May 1999), British Standards Institution,London, 1999.

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttp://www.hindawi.com Volume 2010

RoboticsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Hindawi Publishing Corporation http://www.hindawi.com

Journal ofEngineeringVolume 2014

Submit your manuscripts athttp://www.hindawi.com

VLSI Design

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Modelling & Simulation in EngineeringHindawi Publishing Corporation http://www.hindawi.com Volume 2014

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

DistributedSensor Networks

International Journal of