Residual Strength of Blast Damaged Reinforced Concrete Colu Mns

lable at ScienceDirect

International Journal of Impact Engineering 37 (2010) 295308Contents lists avaiInternational Journal of Impact Engineering

journal homepage: www.elsevier .com/locate/ i j impengResidual strength of blast damaged reinforced concrete columns

Xiaoli Bao, Bing Li*

School of Civil and Environmental Engineering, Nanyang Technological University, 50 Nanyang Ave, 639798, Singaporea r t i c l e i n f o

Article history:Received 1 January 2009Received in revised form30 March 2009Accepted 5 April 2009Available online 23 April 2009

Keywords:Reinforced concrete columnBlast loadingsNumerical simulationResidual axial capacity* Corresponding author. Tel.: 65 67905316.E-mail address: [email protected] (B. Li).

0734-743X/$ see front matter 2009 Elsevier Ltd.doi:10.1016/j.ijimpeng.2009.04.003a b s t r a c t

Columns are the key load-bearing elements in frame structures and exterior columns are probably themost vulnerable structural components to terrorist attacks. Column failure is normally the primary causeof progressive failure in frame structures. A high-fidelity physics-based computer program, LS-DYNAwasutilized in this study to provide numerical simulations of the dynamic responses and residual axialstrength of reinforced concrete columns subjected to short standoff blast conditions. The finite element(FE) model is discussed in detail and verified through correlated experimental studies. An extensiveparametric study was carried out on a series of 12 columns to investigate the effects of transversereinforcement ratio, axial load ratio, longitudinal reinforcement ratio, and column aspect ratio. Thesevarious parameters were incorporated into a proposed formula, capable of estimating the residual axialcapacity ratio based on the mid-height displacement to height ratios.

2009 Elsevier Ltd. All rights reserved.1. Introduction

Columns are the key load-bearing elements in frame structures.Exterior columns are probably the most vulnerable structuralcomponents to terrorist attacks. Column failure is normally theprimary cause of progressive failure in frame structures. However,current knowledge in the evaluation of residual capacity of a blastdamaged reinforced concrete column remains limited. A betterunderstanding of residual capacity in columns would aid in theprediction of the overall performance of buildings, its resistance toprogressive collapse and determining the stability of damagedbuildings especially during search and rescue operations.

Single-degree-of-freedom (SDOF) analysis from blast-resistantdesign guidelines [13] provides engineers with simplifiedanalytical methods to assess blast damage of RC columns. Althoughthese simplified methods are quite useful, three-dimensionalanalysis, in contrast, provides a more in-depth understanding byincorporating all aspects of the response of concrete structuressubjected to blast effects.

A study by Hayes et al [4] suggests that the proper application ofcurrent-practice seismic detailing for high-seismicity regions canreduce vulnerability to blasts and progressive collapse. One of theaims of this study is to quantify this improvement.

A three-dimensional nonlinear FE analysis utilizing the LS-DYNAsoftware [5] is performed for the numerical simulations ofthis research. The FE model is validated through correlatedAll rights reserved.experimental studies. The validated FE model was then analyzedunder simulated blast loads and investigations were carried out onthe dynamic responses and residual axial capacities of the columns.An extensive parametric study was carried out on a series of 12columns to investigate the effect of the transverse reinforcementratio, long-term axial load ratio, longitudinal reinforcement ratio,and column aspect ratio on the column responses.2. Finite element model

The explicit nonlinear FEM program LS-DYNA [5] was utilized inthis study because of its proven effectiveness in geometricmodelling and impact analysis. The description of modellingincludes blast loadings, the structural geometry, relevant materialmodels, application of loads and analysis procedures.2.1. Blast loadings

An exterior explosion to the building generates four types ofloads as shown in Fig. 1: impact of primary fragments, impact ofsecondary fragments, overpressure, and reflected pressure. Thestudy reported within this paper is restricted to the effects ofoverpressure and reflected pressure on the target from an explo-sion. As the overpressure wave strikes on the front face of a closedtarget, reflected pressure is instantly developed, and this is themost destructive aspect of blast loading on a structure. In this study,the explosion centre is assumed at the mid-height of a column,while the surface is assumed to be the reflected surface. The loadingat different points on the front surface of the column for a given

mailto:[email protected]/science/journal/0734743Xhttp://www.elsevier.com/locate/ijimpeng

Fig. 1. Blast loadings on the first floor column during a close-in explosion [6].

X. Bao, B. Li / International Journal of Impact Engineering 37 (2010) 295308296charge and standoff distance is computed by LS-DYNA [5] with thebuilt-in CONWEP blast model, which relates the reflected over-pressure to the scaled distance and also accounts for the angle ofincidence of the blast wave [7]. Blast incidents in recent yearsshowed that most of the terrorist attacks on public structures wereexplosions within short standoff distance (

P

> 0

f'c

f'c/3

f'c

MAX

YIELD

RESIDUAL

MAX

YIELD

RESIDUAL

PT.1

PT.2

PT.3 PT.3: RESIDUALSTRENGTH DUETO CONFINEMENT

PT.2: MAX. STRENGTH

PT.1:YIELD

a b

Fig. 4. Strength model for concrete [8]: (a) failure surfaces in concrete material model; (b) concrete constitutive model.

Fig. 5. (a) DIF for concrete in compression [10]. (b) DIF for concrete in tension [10].

X. Bao, B. Li / International Journal of Impact Engineering 37 (2010) 295308 297where E is the internal energy per initial volume, g is the ratio ofspecific heats. The volumetric strain, 3v; is given by the naturallogarithm of the relative volume. As shown in Fig. 3 the modelcontains an elastic path from the hydrostatic tension cut-off to thepoint T of elastic limit. When tension stress is greater than thehydrostatic tension cut-off, tension failure occurs. When the volu-metric strain exceeds the elastic limit, compaction occurs and theconcrete turns into a granular kind of material. The bulk unloadingmodulus is a function of volumetric strain. Unloading occurs alongthe unloading bulk modulus to the pressure cut-off. Reloadingalways follows the unloading path to the point where unloadingbegan, and continues on the loading path.

A three-curve model is used to analyze the deviatoric stresstensor, as shown in Fig. 4, where the upper curve represents themaximum strength curve, the middle curve is the initial yieldstrength curve and the lower curve is the failed material residualstrength curve.

In order to consider the fact that under higher loading ratesconcrete exhibited increased strength, a dynamic increase factor(DIF), the ratio of the dynamic to static strength, is employed in thisanalysis. The expressions proposed by Malvar and Crawford [10,11]areutilized.TheDIF for theconcrete compressive strength isgivenas:

DIF (

_3=_3s1:026as _3 30 s1

gs_3=_3s

1=3 _3 > 30 s1 (2)where _3 is the strain rate in the range of 30106 to 300 s1;_3s 30 106 s1 (static strain rate); log gs 6:156as2;as 1=5 9fc=fco; fco 10 MPa; fc is the static compressivestrength of concrete. A plot of the formulae employed in thisstudy for the DIF of concrete in compression is shown in Fig. 5(a).

The DIF for concrete in tension is given by:

DIF (

_3=_3sd

_3 1:0 s1

b_3=_3s

1=3 _3 > 1:0 s1 (3)where _3 is the strain rate in the range of 106 s1 to 160 s1;_3s 106 s1 (static strain rate); log b 6d2; d 1=1 8fc=fco;fco 10MPa; fc is the static compressive strength of concrete. A plot ofthe formulae employed in this study for the DIF of concrete incompression is shown in Fig. 5(b).

Based on Fig. 5(a) and (b), it can be seen that the tensile responseismore sensitive to strain rate thancompressive response. Therefore,different rate enhancements are included in tension andcompression in theconcretematerialmodel employed in this study.

Enhanced compressiveMeridian fme

Maximum CompressiveMeridian fm

P

Tensile Meridian

3

f c

fc

rf fc

3

rf fc

fc

ff ft /

use fme (P) = rf * fm (P/rf)

>0

rf ftrf ft

Fig. 6. Rate enhancement in tension and compression [8].

Fig. 7. DIF for reinforcement [10].

Fig. 8. Loading procedures for finite element analysis.

X. Bao, B. Li / International Journal of Impact Engineering 37 (2010) 295308298In the numerical model, the strain rate effect is incorporated asfollows. At any given pressure, the failure surfaces are expanded bya rate enhancement factor which depends on the effective devia-toric strain rate, as shown in Fig. 6. Let rf be the strain rateenhancement factor and p the pressure; an unenhanced pressurep=rf is first obtained, then the unenhanced strength Dsp=rf iscalculated for the specified failure surface. Finally, the enhancedstrength is given by:

Dse rfDsp=rf

(4)

Strength is equally enhanced along any radial stress path,including uniaxial, biaxial and triaxial tension, and uniaxial andbiaxial compression. The effective strain rate versus deviatoricstrength enhancement is given by a LS-DYNA define curve keyword.

2.3.2. ReinforcementSteel is modelled as a strain rate sensitive uniaxial elasto-plastic

material to account for its strain rate sensitivity and stressstrainhistory dependence. The stressstrain curve is assumed to bebilinear, representing an elasto-plastic behaviour with linearisotropic hardening. For the strain rate sensitivity, the expressionsproposed by Malvar and Crawford [10] are utilized. A plot of theproposed formulae is shown in Fig. 7.

The yield stress of reinforcement is represented by:

DIF _3=104

a(5)

where a afy;afy 0:0740:04 fy=414 and fy is the static yieldstrength of reinforcement in MPa.

The ultimate stress of reinforcement is represented by:

DIF _3=104

a(6)

where a afu;afu 0:0190:009 fu=414; and fu is the staticultimate strength of reinforcement in MPa.Eqs. (5) and (6) are valid for reinforcement with yield stressbetween 290 and 710 MPa and for strain rates between 104 s1

and 225 s1.

2.4. Application of loads and analysis procedure

The behaviour of columns subjected to blast conditions will beinfluenced by the fact that in most cases, the columns would havealready been subjected to their respective service prior to beingexposed to blast effects. Therefore, in the first loading stage, gravityload is applied via slow ramps to the column, while in the secondloading stage, blast and gravity loads are applied simultaneously.The gravity load includes both the dead and service loads acting onthe column. The gravity loads are assumed to be in the range of0:10:4f 0cAg to investigate the effects of gravity load on thedynamic response and residual axial capacity of reinforced concretecolumns under short standoff blast conditions. Eventually, in thepost-blast loading stage, axial load is increased until the column iscrushed, as shown in Fig. 8.

3. Verification of finite element model

Verification of the finite element models as outlined in theabove procedure is carried out by performing the analysis of severalcorrelated studies.

3.1. Dynamic testing on concentrically loaded reinforced concretecolumns confined by square hoops

The transverse reinforcement is modelled using beam elementsto take into account the confinement effect. In addition, the

0

50

100

150

200

250

300

350

400

450

500

550

0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

Specific Impulse (psi-sec)

Res

idua

l Def

lect

ion

(mm

)

Laboratory Test Results

Field Test Results

Numerical Results

Fig. 10. Comparison of numerical and experimental residual deflections.

X. Bao, B. Li / International Journal of Impact Engineering 37 (2010) 295308 299concrete material model has one parameter (b1) that governssoftening in compression. The axial loadstrain curve of a smallreinforced concrete column for a uniaxial compression test con-ducted by Mander et al was reproduced [12] for model calibration.

During the experiment, the columnwas loaded concentrically ina DARTEC 10 MN servohydraulically controlled testing machine.Because of the high oil-pumping capacity, a high axial strain ratecould be achieved. For the selected specimen, the strain rate was3c 0:0167=s: The axial strain was recorded over the central400 mm gauge length of column using four linear potentiometers.The axial strain plotted is defined as the average strain occurring inthose gauge lengths within the crushing region rather than theaverage strain of all gauge lengths. Therefore, the critical axialstrain at the central portion of the column from the numericalanalysis is used to match the experimental data. The experimentaland analytical results are plot in Fig. 9. Agreement is close with thepredicted peak axial load exceeding the experimental results byaround 8%.

3.2. Reinforced concrete columns subjected to simulatedblast loading

The first explosive loading laboratory testing program at theUniversity of California, San Diego utilizes a hydraulic-based blastsimulator to simulate explosive events without using explosivematerials [13]. Several tests have been performed to investigate thedynamic response of the reinforced concrete columns when sub-jected to impulsive loads.

The dynamic responses of the test specimens subjected toimpulse loads of 0.77 psi-s, 1.76 psi-s, 1.9 psi-s and 2.3 psi-s wereanalyzed using the proposed finite element models. Three cases ofpositive duration representative of the typical energy dissipationtime of a close-in explosion, 3 ms, 4 ms, and 5 ms, and theirrespective peak pressure were utilized in the analysis, as thedetailed peak pressure and duration for the corresponding impulseloads were not given.

Fig. 10 shows the comparisons of residual deformations ofnumerical, laboratory and field test results. These comparisonsgenerally show a good agreement. The comparisons show that thenumerical result is much higher than the laboratory test resultwhen subjected to an impulse load of 1.76 psi-s. While for the othercases, the comparisons generally show a good agreement. Consid-ering that only limited data are available and the unstable characterof blast test results, these comparisons are considered to be0

2000

4000

6000

8000

10000

0 0.01 0.02 0.03 0.04

Axial Strain (mm/mm)

Experimental

Numerical

Axi

al L

oad

(kN

)

Fig. 9. Comparison of experimental and analytical results.reasonably in good agreement. Good correlations of failure mech-anisms are also observed, as shown in Fig. 11. From both field testand blast simulator test results, it is apparent that the column failedprimarily in diagonal shear near the top and bottom ends, with thecentral portion remaining relatively intact. The predicted damageon the column by the FE model is shown by plotting fringes ofeffective strainwhich is used for measuring the overall deformationat one point. These effective strain contours reveal the strainlocalization where failure propagates. It shows that the failure islocalized near the column top and bottom ends due to diagonalshear failure, which is consistent with field and laboratory testresults.

4. Numerical simulation study

4.1. Numerical simulation matrix

Using the finite element models discussed above, numericalsimulations were performed to evaluate the dynamic response inblast situations, and to further estimate the residual axial capacityof the damaged columns. An extensive parametric study wascarried out with the following cases considered for eachFig. 11. Comparison of numerical and experimental response of reinforced concretecolumns subjected to impulsive loads.

Fig. 12. Column geometry and reinforcement details of the simulation matrix.

X. Bao, B. Li / International Journal of Impact Engineering 37 (2010) 295308300parameter: transverse reinforcement ratio rv 0.12% and 0.46%;longitudinal reinforcement rg 1.8% and 3.2%; column aspectratio L/b 9.8, 8 and 6; axial load ratio PL=f 0cAg 0:1; 0.2, 0.3 and0.4. The parametric study was carried out using a series of 12columns, labelled as series AL, as shown in Fig. 12. The series B,D, F, H, J, and L are seismically detailed columns for which thespacing of the transverse reinforcement is determined in accor-dance with the requirement in the ACI 318 code. Table 1summarises the specimen characteristics of the simulationmatrix.

Fig. 12. (continued).

X. Bao, B. Li / International Journal of Impact Engineering 37 (2010) 295308 3014.2. Numerical results of reinforced concrete columns in thedynamic response stage

4.2.1. Effect of transverse reinforcement ratioBlast loadings are many times greater than conventional loads.

The desired ductile flexural behaviour can only be developed whenthe shear capacity exceeds flexural capacity. The ductility capacityof a column depends on the amount and distribution of transverseTable 1Specimen characteristics of the simulation matrix.

Column type Cross section Column height

A 355 355 mm (14 14 in) 3480 mm (137 in)B 355 355 mm (14 14 in) 3480 mm (137 in)C 355 355 mm (14 14 in) 3480 mm (137 in)D 355 355 mm (14 14 in) 3480 mm (137 in)E 355 355 mm (14 14 in) 2840 mm (112 in)F 355 355 mm (14 14 in) 2840 mm (112 in)G 355 355 mm (14 14 in) 2840 mm (112 in)H 355 355 mm (14 14 in) 2840 mm (112 in)I 355 355 mm (14 14 in) 2130 mm (84 in)J 355 355 mm (14 14 in) 2130 mm (84 in)K 355 355 mm (14 14 in) 2130 mm (84 in)L 355 355 mm (14 14 in) 2130 mm (84 in)reinforcement within the plastic hinge region. The transversereinforcement increases the shear capacity of the column, but moreimportantly, it provides confinement to the core concrete andlateral restraint against buckling of the longitudinal reinforcement.Such restraint is vital for reinforced concrete columns which havebegun to crack and have lost the majority of their tensile andflexural capacity but still need to bear compressive force untilductile failure of the longitudinal reinforcement occurs. Therefore,Aspect ratio Long. steel Transverse steel

9.8 8T25 (rg 3.2%) T10350 mm (rv 0.12%)9.8 8T25 (rg 3.2%) T1088 mm (rv 0.46%)9.8 8T20 (rg 1.8%) T10350 mm (rv 0.12%)9.8 8T20 (rg 1.8%) T1088 mm (rv 0.46%)8 8T25 (rg 3.2%) T10350 mm (rv 0.12%)8 8T25 (rg 3.2%) T1088 mm (rv 0.46%)8 8T20 (rg 1.8%) T10350 mm (rv 0.12%)8 8T20 (rg 1.8%) T1088 mm (rv 0.46%)6 8T25 (rg 3.2%) T10350 mm (rv 0.12%)6 8T25 (rg 3.2%) T1088 mm (rv 0.46%)6 8T20 (rg 1.8%) T10350 mm (rv 0.12%)6 8T20 (rg 1.8%) T1088 mm (rv 0.46%)

Fig. 13. Effect of transverse reinforcement ratio on the displacement response of columns.

0.00

0.02

0.04

0.06

0.08

4500 5000 5500 6000 6500 7000Impulse (kPa-msec)

4500 5000 5500 6000 6500 7000Impulse (kPa-msec)

yr/L

yr/L

g=3.2% v=0.46% L/b=9.8

0.00

0.02

0.04

0.06

0.08

0.10g=1.8% v=0.46% L/b=9.8

P/'c Ag =0.2(D1-D23)

P/'c A

g =0.3(D24-D40)

P/'c A

g =0.4(D41-D52)

P/'c Ag=0.2(B3-B25)

P/'c A

g=0.3(B26-B50)

P/'c A

g=0.4(B51-B67)

a b

Fig. 14. Effect of axial load ratio on the displacement response of columns with a high transverse reinforcement ratio.

X. Bao, B. Li / International Journal of Impact Engineering 37 (2010) 295308302transverse reinforcement is expected to have a significant influenceon the failure mode of the columns under blast loading and theirrespective blast resistance. Fig. 13 shows the effect of the transversereinforcement ratio on the displacement response of the columns.In these figures, the horizontal axis represents themagnitude of theimpulse, which is obtained from the CONWEP software; the verticalaxial yr/L represents the displacement to height ratio, where yr isthe residual lateral displacement at mid-height and L is the clearheight of the column. The dotted lines denote the numerical results,and the solid lines represent the fitted trend lines. Series A and Ccolumns are conventional columns which are mainly designed forgravity loads and are weak in their shear capacity. Series B and Dare seismically detailed columns. It can be seen that under the sameimpulsive loads, the shear-critical columns have a much largerdeflection than the seismically detailed columns. This is consistentwith the predication that implementation of seismic detailing cansignificantly reduce the degree of direct damage due to blast loads0.00

0.01

0.02

0.03

0.04

4300 4400 4500 4600 4700 4800Impulse (kPa-msec)

yr/L

g=3.2% v=0.12% L/b=9.8

P/'c Ag =0.1(A1-A14)

P/'c A

g =0.2(A15-A26)

P/'c A

g =0.3(A27-A30)

a

Fig. 15. Effect of axial load ratio on the displacement responsand consequently improve the blast resistance of the columns. Theimprovement becomes more remarkable in cases with severeimpulsive loads.

4.2.2. Effect of axial loadBefore the occurrence of a blast incident, the gravity load is

already imposed on the column. This will influence the behaviourof the column under blast conditions. Fig. 14 illustrates the effect ofaxial load on the displacement response of the columns with a hightransverse reinforcement ratio. As the figures demonstrate, whenthe impulsive loading and corresponding deformation are small,the mid-height displacement of the column with larger axial loadsis slightly smaller. This is due to the fact that with an increase in theapplied axial load on columns, it would result in an increase in itsmoment capacity and its nominal shear strength. However, thedecrease in mid-height displacement would only occur before theimpulse and its corresponding displacement reach a critical value.4300 4400 4500 4600 4700 4800Impulse (kPa-msec)

yr/L

g=3.2% v=0.12% L/b=8

0.00

0.01

0.02

0.03

P/'c Ag =0.1(E5-E16)

P/'c A

g =0.2(E17-E24)

P/'c A

g =0.3(E25-E29)

b

e of columns with a low transverse reinforcement ratio.

0.00

0.02

0.04

0.06

0.08

0.10

4400 4900 5400 5900 6400 6900Impulse (kPa-msec)

yr/L

v=0.46% L/b=9.8 P/'c Ag=0.2v=0.12% L/b=9.8 P/'c Ag=0.1

g=1.8%(D1-D23) g=1.8%(C1-C11)g=3.2%(A1-A14)g=3.2%(B3-B25)

ab

Fig. 16. Effect of longitudinal reinforcement ratio on the displacement response of columns.

0.00

0.02

0.04

0.06

0.08

4200 4400 4600 4800 5000 5200 5400Impulse (kPa-msec)

yr/L

yr/L

g=3.2% g=3.2% v=0.12% v=0.46%

L/b=9.8(A1-A32)L/b=8.0(E1-E29)L/b=6.0(I1-I7)

0.00

0.02

0.04

0.06

0.08

4500 5000 5500 6000 6500 7000 7500

Impulse (kPa-msec)

L/b=9.8(B1-B67)L/b=8.0(F1-F60)L/b=6.0(J1-J26)

a b

Fig. 17. Effect of column aspect ratio on the displacement response of columns.

Fig. 18. Axial force versus mid-height displacement.

X. Bao, B. Li / International Journal of Impact Engineering 37 (2010) 295308 303Once this critical value is exceeded, the mid-height displacementwould increase greatly with increasing axial load. This is expectedfor columns with flexural behaviour. When columns experiencelarge deflection and plastic hinges formation occurs at mid-spanand fixed ends, axial loads will amplify the lateral deflection andinternal moment due to the PD effect. As the deflection increases,the column will transit from a gradual stiffness and strengthdegradation to a rapid loss of strength due to the buckling of thelongitudinal reinforcement. This explains the change in the slope ofthe curve from gentle to steep as the axial loads increase.

Fig. 15 illustrates the effect of the axial load on the displacementresponse of the columns with a low transverse reinforcement ratio.When impulsive loading and the corresponding deformation aresmall, the mid-height displacement of the columnwith larger axialloads is slightly smaller, as shown in the figure. Columns withhigher axial loads collapsed when the impulse loading and corre-sponding deformation were small due to the brittle behaviour ofshear-critical columns. Thus, there is less data for columns withhigher axial loads shown in the figure. It is believed that axial loadfailure occurred immediately after the shear failure; while for thecolumns having lower axial loads, the collapse only occurred whenthe impulsive loading and corresponding deformation are relativelylarge, although shear failure had occurred at a smaller displacementratio. This may be due to the following factor: the longitudinalreinforcement will support a portion of the axial load up toa maximum load defined by either the buckling (bulking lengthequal to the spacing of ties) or the plastic capacity of the rein-forcement bars; when the axial loads were less than this capacity,the axial load collapse didnt occur until the ductile failure of thereinforcement. Therefore, the columns under low axial loads tendto have a more ductile response than those under high axial loads.Similar findings have been reported for both reinforced concretecolumns and shear walls during seismic investigations [14].

4.2.3. Effect of longitudinal reinforcement ratioAs the longitudinal reinforcement ratio increases, both the

ultimate moment capacity and axial capacity increase. As a result,an increase in longitudinal reinforcement ratio would improve theblast resistance of the column with flexural behaviour. Fig. 16(a)illustrates the effect of the longitudinal reinforcement ratio on the

0.00

0.10

0.20

0.30

0 0.02 0.04 0.06 0.08

yr/L

0 0.02 0.04 0.06 0.08

yr/L

v

L/b=9.8

P/'c Ag =0.1(A1-A13)P/'c Ag =0.2(A15-A26)P/'c Ag =0.3(A27-A30)

P/'c Ag =0.2(B2-B25)P/'c Ag =0.3(B26-B50)P/'c Ag =0.4(B51-B67)

0.00

0.20

0.40

0.60

v

g=3.2% v=0.12% L/b=9.8g=3.2% v=0.46%a b

Fig. 19. Effect of axial load ratio on the ratio of residual axial capacity of the blast damaged columns.

X. Bao, B. Li / International Journal of Impact Engineering 37 (2010) 295308304displacement response of the columns with a high transversereinforcement ratio. The results show that the mid-heightdisplacement decreases with increasing longitudinal reinforce-ment ratio as expected. However, an increase in longitudinal rein-forcement may not always improve the blast resistance. When theflexural capacity of the column exceeds its shear capacity, it maylead to a shift from a ductile flexural failure to a brittle shear failuremode. In such cases, the mid-height displacement will increasewith increasing longitudinal reinforcement ratio. Fig. 16(b) showsthe effect of longitudinal reinforcement for columns with a lowtransverse reinforcement ratio. In this figure, series A columns arecritical in shear due to inadequate transverse reinforcement, andseries C columns are critical in flexure due to the fact that they havelesser flexural capacity than shear capacity because of the lowlongitudinal reinforcement ratio. As the figures demonstrate, whenthe impulsive loading and corresponding deformation are small,the mid-height displacement of series A column is larger than thatof the series C column. This is consistent with the prediction above.However, when the impulsive loading and corresponding defor-mation are large, the mid-height displacement of series C column is0.00

0.10

0.20

0.30

0 0.002 0.004 0.006 0.008

v

L/b=9.8

A1-A30

(yr / L)* (P

L / f'

c A

g)

(yr / L)* (P

L / f'

c A

g)

0.00

0.10

0.20

0.30

0.40

0 0.002 0.004 0.006

v

E1-E29

g=3.2% v=0.12%

L/b=8g=3.2% v=0.12%

a

c

Fig. 20. The nyr=L P=fclarger than that of series A column. This is believed to result fromthe PD effect for flexural columns when the deflection is large.Flexural failure is preferred than shear failure because an extendedplastic response is provided prior to the collapse of column.However, it is noteworthy that the deformation in flexure beyonda certain limit will jeopardize the ability of the column to carryvertical loads due to the PD effect. Therefore, to improve the blastresistance and assure a ductile response, sections need to bedesigned so that the flexural capacity is less than the capacity ofnon-ductile failure mechanism and the deflection must becontrolled to prevent column instability due to the PD effect.

4.2.4. Effect of column aspect ratioIn this study, the cross section is kept constant for all the column

specimens. A decrease in column aspect ratio means a decrease ofoverall height. As a result,when subjected to the same intensityof blastloads, the moment and shear demand will decrease with decreasingcolumn aspect ratio. The effect of aspect ratio on the displacementresponse is shown in Fig. 17. It is observed that under the sameimpulsive loading the displacement to height ratio decreases with(yr / L)* (P

L / f'

c A

g)

(yr / L)* (P

L / f'

c A

g)

0.00

0.20

0.40

0.60

0 0.005 0.01 0.015 0.02 0.025

0 0.005 0.01 0.015 0.02 0.025

v

0.00

0.20

0.40

0.60

v

B1-B67

F1-F60

L/b=9.8g=3.2% v=0.46%

L/b=8g=3.2% v=0.46%

b

d

0Ag curves of columns.

0.00

0.10

0.20

0.30

0 0.002 0.004 0.006 0.008

v

L/b=9.8

(yr / L)* (PL / f c Ag)

(yr / L)* (PL / f c Ag) (yr / L)* (PL / f c Ag)

(yr / L)* (PL / f c Ag)

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0 0.005 0.01 0.015 0.02 0.025

v0.00

0.10

0.20

0.30

0.40

0.50

0.60

v0.00

0.10

0.20

0.30

0.40

0 0.002 0.004 0.006

v

0 0.01 0.02 0.03

v=0.12%

L/b=8v=0.12%

L/b=9.8v=0.46%

L/b=8v=0.46%

g=1.8%(C1-C16)g=3.2%(A1-A30)

g=1.8%(D1-D52)g=3.2%(B1-B67)

g=1.8%(G1-G14)g=3.2%(E1-E29)

g=1.8%(H1-H45)g=3.2%(F1-F60)

a b

c d

Fig. 21. Effect of longitudinal reinforcement ratio on the ratio of residual axial capacity of the blast damaged columns.

X. Bao, B. Li / International Journal of Impact Engineering 37 (2010) 295308 305a reduced aspect ratio as expected. Based on the figures, the results aresimilar for both shear-critical and flexural columns.

4.3. Numerical results of residual axial capacity of the blastdamaged reinforced concrete

In the post-blast analysis stage, the axial load is graduallyincreased by applying a rigid plate attached to the top end of thecolumn using the displacement mode to capture both the residualaxial capacity and the softening portion of the loading curve. Onetypical curve is shown in Fig. 18. It is noted that in the blast loadingstage, due to the inertia effect, the axial load supported by thecolumn is not constant but fluctuates along the deformation of thecolumn. The sudden increase of the axial force at a displacement of75 mm could be due to the steel reinforcing bars reaching theirstrain hardening stage. The increase in the strength of the steelreinforcement at this point in time would result in an increase inthe columns axial load carrying capacity. The ultimate state ofreinforced concrete columns has often been defined by some0.00

0.10

0.20

0.30

0.40

0.50

0.60

0 0.01 0.02 0.03

v

L/b=8

(yr / L)* (PL / f c Ag)

g=3.2%

v=0.12%(E1-E29)g=0.46%(F1-F60)

a

Fig. 22. Effect of transverse reinforcement ratio on the ratioresearchers [15] as a state of vanishing axial capacity to sustain thedead and live loads (long-term load), which is indicated as the pointof axial failure (Fig. 19).

4.3.1. Post-blast damage evaluationIn this study, the residual axial capacity level of a blast damaged

columnwas evaluated by the ratio of residual axial strength, whichwas defined by the following equation:

v PrPL=PmaxPL (7)

where Pmax is the axial capacity of the undamaged columns, Pr isthe residual axial capacity of the damaged columns and PL is thelong-term axial load.

When the column is undamaged, Pr Pmax; the value of v is 1;when the column has lost the ability to sustain the long-term axialload, Pr PL; the value of v just reaches zero, referring as theultimate state of the column. As for the performance indicator, thepreviously defined displacement to height ratio (yr/L) is used.0.00

0.10

0.20

0.30

0.40

0.50

0.60

v

(yr / L)* (PL / f c Ag)

0 0.005 0.01 0.015 0.02 0.025

L/b=8g=1.8%

v=0.12%(G1-G14)v=0.46%(H1-H45)

b

of residual axial capacity of the blast damaged columns.

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0 0.002 0.004 0.006 0.008

v

g=3.2%

L/b=9.8(A1-A30)L/b=8.0(E1-E29)L/b=6.0(I1-I7)

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0 0.01 0.02 0.03

v

L/b=9.8(B1-B67)L/b=8.0(F1-F60)L/b=6.0(J1-J26)

(yr / L)* (PL / f c Ag) (yr / L)* (PL / f c Ag)

v=0.12% g=3.2% v=0.46%a b

Fig. 23. Effect of column aspect ratio on the ratio of residual axial capacity of the blast damaged columns.

X. Bao, B. Li / International Journal of Impact Engineering 37 (2010) 2953083064.3.2. Effect of axial load ratioFig. 19 shows the effect of axial load ratio on the residual axial

capacity of the columns at various degrees of deformation level.The results show that at the same mid-height displacement ratio,the ratio of residual axial capacity is smaller in the case of larger0.00

0.10

0.20

0.30

0 0.002 0.004 0.006 0.008

v

L/b=9.8

Analysis Proposed Equation

Analysis Proposed Equation

Analysis Proposed Equation

0.00

0.05

0.10

0.15

0.20

0.25

0 0.001 0.002 0.003 0.004 0.005

v

0.00

0.10

0.20

0.30

0.40

0 0.002 0.004 0.006

v

(yr / L))* (PL / f c Ag)

(yr / L))* (PL / f c Ag)

(yr / L))* (PL / f c Ag)

g=3.2% v=0.12%

L/b=8.0g=3.2% v=0.12%

L/b=9.8g=1.8% v=0.12%

a

c

e

Fig. 24. Comparison of numerical reaxial loads. As illustrated by the figure, for columnswith lower axialloads, the axial load failure tends to occur at a relatively largedisplacement ratio; for columns with a larger axial load, axial loadfailure tends to occur at a smaller displacement ratio. This indicatesthat the displacement ratio at axial load failure is inversely relatedAnalysis Proposed Equation

Analysis Proposed Equation

Analysis Proposed Equation

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0 0.005 0.01 0.015 0.02 0.025

v

0.00

0.10

0.20

0.30

0.40

0.50

0.60

v

0.00

0.10

0.20

0.30

0.40

0.50

v

0 0.01 0.02 0.03

(yr / L))* (PL / f c Ag)

0 0.005 0.01 0.015 0.02 0.025(y

r / L))* (PL / f c Ag)

(yr / L))* (PL / f c Ag)

L/b=8.0g=3.2% v=0.46%

L/b=9.8g=1.8% v=0.46%

L/b=9.8g=3.2% v=0.46%b

d

f

sults with the proposed curves.

X. Bao, B. Li / International Journal of Impact Engineering 37 (2010) 295308 307to the magnitude of axial load. It is also noted that for columns witha low transverse reinforcement ratio, the effect of axial load is morecritical. This is believed to be due to the significance of confinementduring axial compression.

Based on the observation, a new term yr=L PL=f 0cAg wasintroduced. Fig. 20 plots the relation between v andyr=L PL=f 0cAg for the columns. It can be seen that this new termis capable of reflecting the influence of the axial load ratio on theratio of residual axial capacity. It also highlights the effects of otherparameters and enables them to be studied based on the compar-ison of v versus yr=L PL=f 0cAg curves.

4.3.3. Effect of longitudinal reinforcement ratioIt is usually assumed that longitudinal reinforcement will

support a portion of the axial load up to a maximum load definedby either the buckling or the plastic capacity of the reinforcing bars.In most cases the column collapse is related to the increase of axialload carried by the longitudinal reinforcing bars and their deteri-oration of compressive strength. Fig. 21 shows the effect of thelongitudinal reinforcement ratio on the residual axial capacity ratioof the columns. The results indicate that the ratio of residual axialcapacity is generally larger when the longitudinal reinforcementratio increases.

4.3.4. Effect of transverse reinforcement ratioFig. 22 shows the effect of the transverse reinforcement ratio on

the residual axial capacity of the columns at various degrees ofdeformation level. The results show that the ratio of residual axialcapacity of columns with a low transverse reinforcement ratio issignificantly less than that of the column with a high transversereinforcement ratio. From these figures, it is observed that undersimilar axial loading conditions, axial load failure tends to occur ata relatively large displacement ratio for columns with high trans-verse reinforcement ratio. In contrast, columns with a low trans-verse reinforcement ratio will tend to fail under axial loads ata smaller displacement ratio. This suggests that the displacementratio at axial load failure is directly related to the transverse rein-forcement ratio.

4.3.5. Effect of column aspect ratioFig. 23 illustrates the effect of the column aspect ratio on the

residual axial strength ratio of the columns. The results show that ata high transverse reinforcement ratio, the residual axial capacityratio increases with the reduction in aspect ratio. For columns witha low transverse reinforcement ratio, this effect is not very clear dueto the scatter in the results.4.4. Proposed formulae for determining the residual axial capacityratio of blast damaged reinforced concrete columns

The parametric study carried out revealed the significance of theparameters that affect the residual axial strength of the blastdamaged reinforced concrete column. A formula was derivedthrough multivariable regression analysis in terms of variousparameters to predict the residual axial capacity ratio based on themid-height displacement to height ratio by fitting the parametricstudy results and is as follows:

v h73:65rv 8:465rg0:020879L=b 0:104

ie89284:22rv1308:64221rg9:684203L=b382:12yr=LPL=f 0cAg (8)

A few examples presenting the comparison of the proposedequation with the analytical results are shown in Fig. 24. The solidline in each plot denotes the proposed equation, while the dottedline represents the numerical analysis results. It is observed thatfor most parts, the proposed curves are close to the analyticalresults.5. Summary and conclusions

Based on the results of the parametric study, the following mainconclusions can be drawn.

The numerical results show that the use of seismic detailingtechniques can significantly reduce the degree of direct blast-induced damage and subsequent collapse of the reinforcedconcrete columns.

Comparisons of the deterioration of the axial strength underdifferent axial load ratios indicate that the ratio of residual axialstrength is smaller under larger long-term axial load. The effectof axial load ratio is more critical in the case of columns witha low transverse reinforcement ratio.

The results indicate that the ratio of residual axial capacitygenerally increases with an increase in the longitudinal rein-forcement ratio.

The numerical results indicate that the residual axial capacityratio increases with a reduction in the aspect ratio for columnswith a high transverse reinforcement ratio.

A formula was derived by fitting the parametric study results,in terms of various parameters to predict the residual axialcapacity ratio based on the mid-height displacement to heightratio. The comparison of the proposed equation with theanalytical results shows that the proposed curve well repre-sents the tendency with the variation of the parameters. Futureexperimental investigation of residual axial strength of theblast damaged reinforced concrete columns is needed tofurther supplement the limited data set used to develop theproposed equation.

Acknowledgements

This research was supported by a research grant provided by theDefense Science and Technology Agency (DSTA), Singapore, underthe Protective Technology Research Center, Nanyang TechnologicalUniversity, Singapore. Any opinions, findings and conclusionsexpressed in this paper are those of the writers and do not neces-sarily reflect the view of DSTA, Singapore.

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Residual strength of blast damaged reinforced concrete columnsIntroductionFinite element modelBlast loadingsStructural geometry modellingMaterial modelsConcreteReinforcement

Application of loads and analysis procedure

Verification of finite element modelDynamic testing on concentrically loaded reinforced concrete columns confined by square hoopsReinforced concrete columns subjected to simulatedblast loading

Numerical simulation studyNumerical simulation matrixNumerical results of reinforced concrete columns in the dynamic response stageEffect of transverse reinforcement ratioEffect of axial loadEffect of longitudinal reinforcement ratioEffect of column aspect ratio

Numerical results of residual axial capacity of the blast damaged reinforced concretePost-blast damage evaluationEffect of axial load ratioEffect of longitudinal reinforcement ratioEffect of transverse reinforcement ratioEffect of column aspect ratio

Proposed formulae for determining the residual axial capacity ratio of blast damaged reinforced concrete columns

Summary and conclusionsAcknowledgementsReferences