ar

W2

y-dndcaply-ardve svesstitvolution. The pressureimpulse diagrams for both in-plane and out-of-plane blast

been arch thuake loxceptirgy dis

t qualities of conven-ith the additional goalquired for optimalved through the devel-

Journal of Constructional Steel Research 106 (2015) 4456

Contents lists available at ScienceDirect

Journal of Constructioexible than concrete construction in terms of future expansion andsite rearrangement. Therefore, having a reliable protective structuralsteel option available would be advantageous economically. Through

opment of pressureimpulse (PI) diagrams. First, the PI curve isdescribed in detail and is generalized (normalized) by transforming awall system into a single-degree-of-freedom (SDOF) system. A compre-potential applications as protective structures have received littleattention.

Most of the structures in industrial plants are made up of steel sys-tems, which normally have rapid erection times and tend to be more

This research is an exploration of the inherentional SPSWs for use as protective structures, wof identifying where modications are reperformance in this newapplication. This is achieassigned it the highest ductility-related and overstrength-related forcemodication factors of any seismic system. Although it is undeniablywell-suited for high seismic regions, and similar properties are advanta-geous for resisting other types of dynamic loading such as blast, their

of alternative load paths. As such, the SPSW, which is a continuoussystem with a high level of ductility capacity, is potentially a good can-didate as a primary component of protective structures in industrialplants.the process of site planning, protective strucare sited at a suitable distance from process eq

Corresponding author. Tel.: +1 780 492 0091; fax: +E-mail address: rdriver@ualberta.ca (R.G. Driver).

http://dx.doi.org/10.1016/j.jcsr.2014.11.0100143-974X/ 2014 Elsevier Ltd. All rights reserved.sipation capacitywithoutstem has reached a stageStandard S16 [1], have

compromised if they are to fulll their intended function. To limit thedamage and improve the reliability of the system, a high level of redun-dancy is benecial for blast-resistant systems to ensure the availabilitydegrading under cyclic loading. As such, the sywhere design standards, such as Canadian1. Introduction

Steel plate shear walls (SPSW) havetwo decades primarily based on reseatheir performance under severe earthqhas shown that the system possesses eforce resistance, with a high level of enethe numerical model. Different response criteria and wall sizes are considered. A method is proposed to producedimensionless iso-response curves to broaden their applicability. The results show that despite the inherent slen-derness of the steel members, the wall system has the potential to be an effective system for use in a protectivestructure for industrial plants, especially for the in-plane blast load condition.

2014 Elsevier Ltd. All rights reserved.

dvanced through the lastat focuses on improvingading. Previous researchonal ductility and lateral

of release of ammable or explosive material. As such, the blast loadsthat need to be considered in the design of industrial structures tendto be accidental far-range (low pressure) detonations and are lessdetrimental for slender steel members than near-range (high pressure)explosions. Protective structures are prone to localized damage andfailures under blast loading, but their overall integrity must not bePerformance assessment orientations, along with the corresponding weightstandoff distance diagrams, are produced with the aid ofNumerical methodsBlast loading

model is developed. The conand damage initiation and ePerformance assessment of steel plate sheblast loads

Hassan Moghimi, Robert G. Driver Department of Civil and Environmental Engineering, University of Alberta, Edmonton, AB T6G 2

a b s t r a c ta r t i c l e i n f o

Article history:Received 11 February 2014Accepted 28 November 2014Available online 23 December 2014

Keywords:PI diagramSteel plate shear wall

Previous research on properlshear strength and stiffness asuperior energy dissipationresisting system for seismicaltheir application in this regaplate shear wall as a protectiby means of iso-response curtures in industrial plantsuipment and any source

1 780 492 0249.walls under accidental

, Canada

esigned steel plate shearwall systems has proven that it has a high level of laterallarge ductility. These properties, along with ample redundancy, robustness, andacity under severe cyclic loading, have made the system a viable lateral forcective regions. Although similar properties are desirable for protective structures,has been largely neglected. The potential application of some form of the steelystem in industrial plants possibly subjected to accidental explosions is studied. To capture all important aspects in blast response, a comprehensive numericalutive model for the steel material includes mixed-hardening, strain-rate effects,

nal Steel Researchhensive numerical model that is able to capture all critical aspects of theblast response is developed. The in-plane and out-of-plane responsesare investigated separately. PI diagrams for two different-size wallshave been developed and normalized. They are then converted tocharge weightstandoff distance curves. The results show that a

properly-designed and detailed SPSWmay indeed be a viable protectivesystem for accidental blast in industrial plants such as petrochemicalfacilities.

2. Literature review

2.1. Performance criteria

The maximum dynamic responses of structural componentsintended to resist the blast loading need to be limited against the de-sired blast levels of protection or blast design objectives. These responselimits are typically called performance criteria, and are dened in blastdesign guidelines. Generally,when components are under large shear orcompressive forces, the response limits are small and barely reach theyield point, while large deformation limit values are permitted forcomponents loaded mainly in exure. Additionally, other factors, suchas the siting distance from the blast source, occupancy of the building,

and VL represent High, Medium, Low, and Very Low levels of protec-tion, respectively. The column Resp. Param. shows the different re-sponse parameters at each level of protection, which are the ductilityratio, , and support chord rotation, (). Because of different denitionsof damage or performance level in the various design guides, directcomparisons of the response limits shownmaynot be absolutely consis-tent; the table is intended for general comparisons only.

For the response limits proposed by the ASCE blast design manual[2], the performance levels are identied in table column Perf. Level,as Low Resp., Med. Resp., and High Resp., which correspond withthe high, medium, and low levels of protection, respectively. Themanual provides dened response limits for different hot-rolled steelcomponents, including compact secondary exural members such asbeams, girts, and purlins (column BM Sec.), primary frame memberswith and without signicant compression (column Prim. Mem.), andplates (column PL). Signicant compression is dened as a force largerthan 20% of the dynamic axial compressive capacity of the member,

(20

p. d

rc

erat

vy

rdo

45H. Moghimi, R.G. Driver / Journal of Constructional Steel Research 106 (2015) 4456and importance of the equipment protected by the building, affect theblast design requirements.

The American Society of Civil Engineers' (ASCE) document for blastdesign of petrochemical facilities [2] denes the allowable deformationof individual components based on the desired level of protection andtype of component for different construction material types. Threeperformance levels, or damage levelsnamely, low, moderate, andhigh response rangeshave been considered. The performance levelsare conceptually similar to the immediate occupancy, life safety, andcollapse prevention performance levels, respectively, used inperformance-based seismic design [3]. The low response range corre-sponds to a high degree of blast protection with only localized damage.The medium and high response ranges represent widespread damageand loss of structural integrity, respectively. Two dimensionless re-sponse parametersnamely, the ductility ratio, , and support (chord)rotation in degrees, have been dened at each performance level.The ductility ratio is the ratio of the maximum component deformationto its yield deformation,which is ameasure of the capability of the com-ponent to experience inelastic deformation and absorb energy with nosignicant capacity loss. The tangent support rotation is a measure ofboth rotational ductility at the support and the degree of potential insta-bility in the member. Building performance criteria are also dened inthe ASCE document according to the inter-story drift ratio. For example,the lateral drift ratios of moment-resisting structural steel frames arelimited to 2.0%, 2.85%, and 4.0% for the low, medium, and high responseranges, respectively. The response limits have been elaborated from therst edition of the document published in 1997, and the changes havebeen described in detail by Oswald [4].

The response limit for an individual structural steel componentbased on various design guidelines are shown in Table 1. The columnLP shows the component Level of Protection, where H, M, L,

Table 1Response limits for hot-rolled structural steel members in various blast design guides.

LP Resp.param.

ASCE (2011) [2] UFC (2008) [5] PDC[7]

Perf. level BM sec. Prim. mem.a PL Prot. cat. BM PL Com

H Supe Low Resp. 3 1.5 (1.5) 5 2 1 (1) 3

M Med. Resp. 10 3 (2) 10 Cat. 1 10 10 Mod 6 2 (1.5) 6 2 2

L High Resp. 20 6 (3) 20 Cat. 2 20 20 Hea 12 4 (2) 12 12 12

VL Haza

a For member with signicant compression, the values in parentheses should be usedb For combined exure and compression, the values in parentheses should be usedwhere the axial force is evaluated from a capacity method based onthe ultimate resistance of the supported members exposed to theblast loads.

TheUFC 3-340-02 document [5] presentsmethods of design for pro-tective construction against accidental explosion of high-explosive(mainly military) materials. Two levels of protection have been consid-ered for blast design. Structures designed to protect personnel againstaccidental blast are classied as Category 1, while structures providedto protect equipment are designated Category 2. The response criteriaproposed by this document are shown in Table 1, where the columnProt. Cat. shows the protection categories, and columns BM andPL show the response limits for beams and plates, respectively.Categories 1 and 2 (Cat. 1 and Cat. 2, respectively, in the table)correspond to medium and low levels of protection, respectively.

The PDC TR-06-08 document [6] denes response criteria againstexplosive terrorist threats in terms of ductility ratio and support rota-tion for four different component damage levels, including Supercial,Moderate, Heavy, and Hazardous, as shown in Table 1. Different struc-tural component types and characteristics have been considered forboth primary and secondary elements. Table column Comp. Dam.shows the component damage levels, where columns BM, CPR,and PL show response limits for primary compact beam elements,compression members, and plates, respectively. (The document alsosuggests response limits for non-compact, secondary, and non-structural components, not shown in the table.) The moderate andheavy component damage levels correspond roughly to the mediumand high response performance levels, respectively, in the ASCE petro-chemical design guidelines [2]. However, the supercial damage levelrepresents more conservative design limits (i.e., lighter damage levels)than the low response performance level in the ASCE manual [2], butboth can be classied as high levels of protection. Since the response

08) [6], ASCE/SEI (2011) ASCE (1999) [8] NYC (2008)[9]

am. BMb CPR PL Dam. level BM ex. BM shr. CPR BM sec. CPR

ial 1 (1) 0.9 4 () 1 Light dam. 10 5.7 2.3

e 3 (3) 1.3 8 Mod. dam. 20 3 (3) 2 13.5 4.6

12 (3) 2 20 Severe dam. 40 20 510 (3) 6 26.6 9.1 10 6

us 25 (3) 3 40 20 (3) 12

46 H. Moghimi, R.G. Driver / Journal of Constructional Steel Research 106 (2015) 4456limits have been developed primarily based on static test data, they aregenerally more conservative in comparison with other criteria [4]. Inorder to enable the user to assess the overall building blast protection,the PDC document [6] also denes the hazardous damage level, corre-sponding to a very low level of protection, although it is not generallyselected for blast design. The ASCE/SEI blast design document [7] hasadopted the same response limits as PDC TR-06-08 [6].

The ASCE document on structural design for physical security [8] de-nes the response limits for light, moderate, and severe damage, and inTable 1 in the column Dam. Level these are indicated as Light Dam.,Mod. Dam., and Severe Dam., respectively. The columns BM Flex.and BM Shr. describe response limits for beams subjected to exur-al/membrane and shear actions, respectively, and column CPR pro-vides the limits for columns under compression. Since the responselimit values in this document are presented in such a way that theyare not directly comparable with those in the other manuals, they aretransformed accordingly as follows. The beam response limits havebeen dened in the document in terms of the ratio of the centerlinedeection to the span (/L). Assuming a symmetric response, thevalue 2/L is the tangent of the support rotations. Therefore, by doublingthe limits and taking the inverse tangent, the support rotation is calcu-lated and shown in Table 1. Also, the ratio of shortening of the column toits original height is provided for the compression-column responselimit. This ratio is equal to the average compressive strain in the column.Assuming the application of steel with a yield stress of 400 MPa, theyield strain would be 0.2%. By dividing the column limit ratios by 0.2%,the strain ductility ratio is calculated and presented in the table. Com-paredwith other design guides, this document suggests larger responselimit values because they represent structural damage levels observedin experimental/numerical results. As such, these values are suggestedfor post-event assessment and not for design.

The response criteria of the New York City building code [9] are alsoshown in Table 1. This code provides dened response limits for steelbeams (column BM Sec. in the table) and steel columns or compres-sion members (column CPR). The criteria are dened for behavior ofa single element and the rotation criteria refer to the support rotation.Although just one response limit is dened, it is close to the ASCE [2]criteria for the low level of protection, which suggests a design at thecollapse prevention performance level.

2.2. PI curves

Abrahamson and Lindberg [10] illustrated the characteristics of PIcurves for elastic and rigid-plastic undamped SDOF systems subjectedto idealized rectangular, triangular, and exponential-shaped pulseloads with zero rise times. The results showed that the pulse shapehas virtually no effect on the impulsive and quasi-static loading regimes,but it changes the response in the dynamic loading realm. Iso-damagecurves were produced for uniformly-loaded, rigidplastic beams andplates, and for dynamic buckling of cylindrical shells under uniformlateral loads.

Baker et al. [11] describe three response regimes for undamped lin-ear elastic SDOF systems subjected to an exponentially decaying load,and the results were compared with an analogous rigidplastic system.The PI curves were produced for non-ideal explosions applied to anundamped elastic SDOF system. Two different cases for blast waveswith nite rise time (e.g., conned gas or dust explosion) and blastwaves with zero rise times and a negative phase (e.g., pressure vesselburst) were studied. The impulsive and quasi-static asymptotes weredeveloped for different systems by means of energy solutions. It wasshown that the dynamic response associated with any iso-responsecurve depends only on the maximum load and system stiffness on thequasi-static loading asymptote and only on the impulse and systemmass and stiffness on the impulsive loading asymptote. In the lattercase, the authors claim that any peak load and duration combination

with equal impulse results in the same dynamic response.Krauthammer et al. [12] describe three different search algorithmsto derive PI diagrams numerically. The methods are compared withclosed-form solutions for undamped linear elastic SDOF systems sub-jected to rectangular, triangular, and exponential pulse loading typeswith instantaneous rise times, and good agreement was reported.

2.3. SPSW systems

SPSW systems have been the subject of very little blast research inthe past. The out-of-plane blast resistances of two 40%-scale single-story SPSW specimens designed for seismic loads were studied byWarn and Bruneau [13]. The rst specimen was subjected to a smallcharge weight in close proximity, with a scaled distance of Z (specicvalues were not reported). After the test, signicant residual inelasticdeformations were observed in the inll plate, but it remained attachedto the boundary frame. The base beam sustained larger inelastic defor-mations than the top beam, with cracks forming in theweld connectingthe top ange of the base beam to the column anges. The second spec-imen was subjected to a larger charge with a longer standoff distancecompared with the rst specimen, such that the scaled distance was1.66Z. Although this specimenwas subjected to a smaller blast load ef-fect due to the larger scaled distance, the inll plate unzipped aroundthree sides of the panel because of the failure of the weld connectingthe inll plate to the sh plates. The out-of-plane resistance of thewall was estimated using yield line theory and an approximate plasticanalysis method. The former method underestimates the out-of-planeresistance of the wall, while the latter one overestimates the strengthof the system.

Moghimi and Driver [14] studied the overall performance of a SPSWsubjected to in-plane and out-of-plane blast load using numericalmethods. The blast loads were shock waves with an appropriateduration for the design of petrochemical facilities. Local and global dam-age indices were suggested for the system. Dynamic performances ofthe system in both blast load directions were studied. Blast resistancecapacities were assessed according to both total absorbed strain energyand maximum structural displacement.

3. PI diagrams

3.1. PI diagram versus response spectrum curve

For designing systems subjected to dynamic loading, the maximumresponse, rather than the response time history, is normally of mostinterest. As such, response spectra or dynamic load factorsthe ratio ofmaximum dynamic to static response as a function of dynamic load du-rationhave been used in dynamic design of structures for several de-cades. As shown by Baker et al. [11], the typical response spectrumdiagram for an undamped linear SDOF system subjected to an exponen-tially decaying pulse loadwith instantaneous rise time and innite dura-tion would be similar to the one depicted in Fig. 1(a). The system has amass and stiffness of m and k, respectively, with a resulting naturalangular frequency ofn=(k/m)= 2/Tn, where Tn is the natural peri-od of vibration of the system. As shown in Fig. 1(a), the load, which rep-resents an air blastwave, is dened by the equation p(t)= P0 exp (t/T),and the impulse of the load would be I= P0 T, where T can be describedas an equivalent loading duration for a rectangular pulse with the samepeak value as the actual decaying load. For the described system, the or-dinate of the response spectrum curve is the ratio of maximum dynamicdisplacement (Xmax) to the maximum static displacement (xs = P0/k),which is y = Xmax/(P0/k), and the abscissa is the scaled time, or theratio of equivalent loading duration to the natural period of vibration,which (scaling by the constant 2 for convenience) is x=2T/Tn=n T.

The response spectrum curve shown in Fig. 1(a) has two asymp-totes. The rst is an inclined asymptote (y = x) that passes throughthe originwith the slope=1and the second is a horizontal asymptote

(y= ) with a y-intercept of =2. As described by Baker et al. [11], the

dynamic load (P) and associated impulse (I)that produce the same

d, (b) Nondimensional mapped PI diagram, (c) Ordinate transformation (y to Y), (d) Abscissa

47H. Moghimi, R.G. Driver / Journal of Constructional Steel Research 106 (2015) 4456diagram consists of three loading regimes: impulsive (x= n T b 0.4),dynamic (0.4 x = n T 40), and quasi-static (x = n T N 40). Thecurve provides useful information in the dynamic and quasi-static do-mains, while it suppresses the system response in the impulsive loadingregimewhere the duration of pulse load is short relative to the responsetime of the system (i.e., as x converges to zero).

Since blast design deals mostly with the impulsive loading regime,the response spectrum is normally not an appropriate tool. As such,the response spectrum needs to be transformed into a different coordi-nate system for use in blast design, which is called a peak load (or peakpressure)impulse (PI) diagram. A typical PI curve is shown inFig. 1(b), which has horizontal and vertical asymptotes, both withunity intercepts. Two mappings are required to derive Fig. 1(b) fromFig. 1(a), and Figs. 1(c) and (d) show the ordinate and abscissatransformations, respectively. The ordinate is transformed such thatthe impulsive loading regime response,which passes through the originin the original response spectrum curve, is retrieved. While the trans-formation 1/y does the job, in order to have the horizontal (quasi-staticloading regime) asymptote with a unity intercept, the transformationY= (1/y) , as shown in Fig. 1(c), can be used. As dened in Fig. 1(b),the new ordinate, Y = P0/[(kXmax)/2)], is a normalized or nondimen-sional load for a linear SDOF system where the numerator is themaximum pulse load and the denominator is the average equivalentelastic static force due to the maximum dynamic deformation.

Fig. 1(c) still has an inherent restriction, since when Y (the dimen-sionless load) is a large value in the impulsive loading regime, x (theperiod ratio) converges to zero for all systems and applied pulse loads.To change this characteristic of the curve, the inclined asymptote in theresponse spectrum curve in Fig. 1(a) is mapped to a vertical asymptote.Since the original response spectrum curve has a linear inclined asymp-tote, the mapping x/y for the abscissa brings the inclined asymptoteto a vertical asymptote with 1/ intercept. As such, X = (x/y) mapsthe inclined asymptote to the vertical (impulsive loading regime) one

Fig. 1. (a) Typical response spectrum curve for SDOF under exponentially decaying pulse loatransformation (x to X).with a unity intercept, as shown in Fig. 1(d). As dened in Fig. 1(b), thenew abscissa, X= I/[(Xmaxn)m], is a normalized or nondimensional im-pulse, where the numerator is the impulse, or area under the pulse load,and the denominator is an equivalent impulse or change in momentumper unit time for an elastic SDOF system under free vibration with a dis-placement amplitude of Xmax.

The PI and response spectrum curves encompass the same infor-mation and are essentially different representations of the same re-sponse. However, they emphasize different dynamic responsefeatures. The response spectrum curve displays the (nondimensional)dynamic response as a function of the (nondimensional) dynamic loadduration. In otherwords, it shows how the dynamic response is affectedby the load duration. The curve has two linear asymptotes. The quasi-static loading (horizontal) asymptote shows that any change indynamic load duration within this regime does not have any effect onthe maximum dynamic response. The PI curve, on the other hand, isthe locus of equivalent dynamic loadsa combination of maximummaximum dynamic response in the system. In other words, for anyselected response limit in the system, the curve represents an iso-response as a function ofmaximumdynamic load and its correspondingimpulse. The PI curve has both linear vertical (impulsive) and horizon-tal (quasi-static) asymptotes. The critical parameter in the impulsiveloading regime is the impulse value. That is, any change in just themax-imum loadmay not cause the response to deviate from the iso-responsecurve, while a change in only the impulse necessarily does. In the sameway, the critical parameter in the quasi-static loading regime asymptoteis the maximum load.

Historically, PI curves have been developed for damage assessmentof structures under severe blast and transient dynamic loads to denethe peak load and impulse combinations that result in a specic damagelevel. As such, they are commonly known as iso-damage curves. Howev-er, in this research the terms PI curve and iso-response curve are inter-changeably used, instead of iso-damage curve, since they better conveythe concept of the curve.

3.2. Trial-and-error approach

A PI curve can be developed for a simple linear elastic SDOF systemsubjected to well-dened dynamic loads by analytical methods orenergy conservation principles. The former method, as represented byFig. 1(b), can even result in a closed-form solution for simple cases,while the latter method is mainly used to derive the solution in the as-ymptotic regions [11]. However, both methods are applicable to simpleand well-dened cases. The numerical solution is the only reasonablemeans of deriving the iso-response curves for a general nonlinearsystem subjected to complex loading functions.

The PI curve corresponding to a given SPSW and a selectedresponse limit is generated by curve-tting on a sufcient number ofFig. 2. Trial-and error method to nd single points on iso-response curve.

48 H. Moghimi, R.G. Driver / Journal of Constructional Steel Research 106 (2015) 4456points computed by numerical analysis. Each point on the curve isrecovered by a series of numerical analyses of the system and a trial-and-error method [12]. Fig. 2 demonstrates the method to nd twoarbitrary points (Ii,Pi) and (Iq,Pq) on the impulsive and quasi-staticasymptotes, respectively. By keeping the pressure Pi constant andchanging the impulse (Ii,j), where the index j represents the number ofthe iteration that varies from 0 to n, the threshold impulse in the impul-sive asymptote that satises the response limit (Ii,n = Ii) is found. Simi-larly, for the quasi-static asymptote, the impulse Iq is kept constant andthe pressure (Pq,j) changes until it satises the response limit criterion(Pq,n= Pq). As such, each point (Ii,j,Pi) or (Iq,Pq,j) in the gure representsthe results from a single dynamic analysis. Among them, the two points(Ii,Pi) and (Iq,Pq) belong to the iso-response curve and they indicate twocombinations of impulse and peak pressure that cause the maximumdynamic response of the system to just reach the selected responselimit. In the dynamic response domain, either of the methods men-tioned (i.e., constant peak pressure or constant impulse) can beimplemented.

3.3. Nondimensional PI diagram

Although iso-response curves are powerful and provide an effectiveblast design tool, they are inherently tied to themechanical and dynam-ic properties of the system under consideration. Therefore, for any givenresponse limit, a new PI curve should be developed if the system ismodied, which requires many nonlinear nite element analyses orblast tests. Since each PI curve deals with one specic response, thesystem can be transformed into an equivalent single-degree-of-freedom (ESDOF) system for the degree-of-freedom (DOF) associatedwith the response limit [15]. With ESDOF system properties, a PIcurve developed for a specic target response and SPSW system canbe normalized and used for different walls with similar properties.

3.3.1. Nondimensional load/pressureConsidering an elasto-plastic SDOF system under a dynamic load

with an amplitude of F, the external work is equal to F m and the strainenergy is equal to Vy (m 0.5 y)= 0.5 Vy y (2 1), where y and Vyare the yield displacement and yield resistance (elastic limit) of thesystem with an elastic stiffness k= Vy/y, m is the maximum displace-ment of the system, and = m/y is ductility ratio.

The ideal dynamic load (with an instantaneous rise time withinnite duration), P0, which produces the maximum displacement m0or ductility 0 in the system can be calculated using conservation ofenergy:

P0 0:5 Vy 21=0 1

P0would be the quasi-static asymptote of the iso-response curve for themaximum response of m0 = 0 y in the elasto-plastic SDOF system.The blast load applied to the system can be normalized with such aload. However, Eq. (1) is valid only for a SDOF system. For a multi-degree-of-freedom (MDOF) system such as a SPSW, it must be trans-formed into an ESDOF system through transformation factors [15]. Assuch, considering Eq. (1), the following equation for the nondimension-al blast load on the system is used:

Pe=Pe0 KL P=KL 0:5 Vy 21=0 P=0:5 Vy 21=0 h i

P=P0 2

where the subscript e is the equivalent value for the distributed system(realwall), P is the resultant force of the total applied blast pressure, andKL is the load transformation factor. At the verge of yielding (0=1), thestrain energy would be equal to 0.5 k y2, the ideal force that producesthe yield displacement in the system would be equal to P0 = 0.5 Vy,

and the nondimensional force would be P/P0 (Eq. (2)).3.3.2. Nondimensional impulseThe ideal impulse (with zero duration), I0, that produces the yield

displacement in an elasto-plastic SDOF system can be found by usingconservation of energy. By equating the above-dened strain energywith the kinetic energy, which is equal to I02/(2m) in the elastic regionfor a SDOF system with mass m:

I0 k m y 201 m y 201 Vy m y

201 Vy=

201 3

where = (k/m) is the natural frequency of the system. I0 would bethe impulsive asymptote of the iso-response curve for themaximum re-sponse of m0= 0 y. Similarly, Eq. (3) is for a SDOF system, and for thedistributed system the transformation factors should be applied totransform it into an ESDOF system:

Ie0 ke me y 201 KM KL k m y 201 KM KL I0 4

where KM is themass transformation factor. Also, the equivalent appliedimpulse would be equal to:

Ie 0:5 Pe td KL 0:5 P td KL I 5

where td is the triangular pulse load duration. The nondimensionalimpulse is (using Eqs. (4) and (5)):

Ie=Ie0 KL 0:5 P td = KMKL I0 KL I= KMKL I0

KL=KM I=I0 1= KLM I=I0 6

where I is the total impulse of the blast load and KLM = KM/KL is theloadmass transformation factor [15]. At the end of the elastic region(0 = 1), the strain energy would be equal to I02/(2 m) and the idealimpulse that produces the yield displacement in the system would beI0 = (k m) yfor a MDOF system it would be Ie0 = (KM KL) I0andthe nondimensional impulse would be equal to Ie/Ie0 = 1/(KLM) I/I0(Eq. (6)).

Eqs. (2) and (6) dene the parameters for the nondimensional PIcurves.

4. Material model for blast analysis of SPSWs

To study the performance of the SPSW systemunder accidental blastloading, a previously-tested SPSWdesigned for seismic applications hasbeen selected. The system is a half-scale four-story SPSW tested byDriv-er et al. [16] under vertical gravity load concurrent with cyclic lateralloads resembling the effect of a seismic event. The commercialgeneral-purpose nite element code ABAQUS is used to model the testspecimen. The material plasticity was represented by the von Misesyield criterion along with isotropic hardening for monotonic loadingand kinematic hardening for cyclic loading, and the models werevalidated by Moghimi and Driver [14].

To develop a reliable PI diagram from numerical study, a compre-hensivenite elementmodel capable of considering all important issuesthat affect the blast response is required. As such, the numerical modelby Moghimi and Driver [14] is enhanced further in the current study.The welds connecting the inll plate to the surrounding frame aremodeled explicitly, using weld material properties based on the test re-sults of transversely-loaded llet welds by Ng et al. [17]. The selectedyield and ultimate stress values are 470 and 630 MPa, respectively,while the strain corresponding to the ultimate stress and fracture strainare selected as 0.10 and 0.22, respectively. The weld material has ahigher strength with lower ductility than the structural steel used forthe inll plates and boundary frame.Moreover, the enhancedmodel in-cludes a comprehensive hardening rule, stress dependency on strain

rate, and damage accumulation in the steel materials, which are

important properties to arrive at reliable iso-response curves. Theseproperties are described in detail in the following section; however,the material model needs further experimental verication.

4.1. Constitutive material model

4.1.1. Hardening rulesSince the blast response of SPSWs involves yielding and cyclic re-

sponse, a suitable hardeningmodel is required for achieving acceptable

49H. Moghimi, R.G. Driver / Journal of Constructional Steel Research 106 (2015) 4456results. The isotropic hardening rule assumes that as plastic deforma-tion develops, the subsequent yield surface experiences no translationand just expand its size symmetrically in the stress space from its initialyield surface. As such, it is most appropriate for a monotonic loadingcondition. The kinematic hardening model assumes that as the plasticdeformation develops, the subsequent yielding surface experiencesrigid body translation with no change in the size of the yield surface.Although the kinematic hardening rule may lead to acceptable resultsfor structures subjected to mild dynamic loading, it imposes twomajor limitations. First, the size and orientation of the initial yield sur-face remains unchanged. Second, it requires a modied bilinearstressstrain curve, which does not allow strain softening to occur iflarge strains are experienced in thematerial subjected to the blast load.

In this study, mixed hardening with full stressstrain properties, in-cluding softening, is used. This is in effect a combination of the isotropicand kinematic hardening rules. As plastic deformation advances, theyield surface can experience translation, while at the same timeallowing growth in its size in the stress space. In the current study, a10% increase in the size of the yield surface after initial yield is assumed,which is believed to be conservative (although the results are not partic-ularly sensitive to this quantity), and the remaining increase in the yieldstress is allocated to the translation of the yield surface.

4.1.2. Rate dependent yieldThe high strain rate in the material resulting from the blast load

causes an increase in the yield stress level compared to that in a staticloading regime. Neglecting the strain rate effect results in a conservativeexural design for the components directly subjected to the blast load.However, it underestimates the induced design forces from the exuralaction in the component, such as the design shear force for thememberitself or the design forces on its connections and supporting members.As a result, the rate dependency of the material strength is consideredin the current study, since it could have a non-conservative design ef-fect, especially in the out-of-plane blast direction. The suggested valuesin the UFC document [5] for the yield stress increase for structural steelare presented in Table 2. Different values are proposed for normal andhigh strength steels. As shown in the table, values slightly closer tothose for A36 steel than A514 steel are selected in the current study toreect the properties of the steel used in the test. Although the increasein the yield stress is larger than the increase in theultimate stress, due tolack of information on the latter, the entire post-yield stressstraincurve is increased by the values in Table 2. The effect of this approxima-tion is considered negligible, however, since the strain values that de-velop in the system are generally much less than the ultimate strainand barely pass the yield plateau.

Table 2Strain rate effects on strength increase in structural steel.

Strain rate(mm/mm/s)

Strength increase factor

ASTM A36a ASTM A514b Average Selected value

0.1 1.29 1.09 1.19 1.21 1.45 1.17 1.31 1.410 1.6 1.27 1.44 1.5

a Minimum yield stress of 250 MPa and ultimate tensile strength of 400550 MPab Minimum yield stress of 690 MPa and ultimate tensile strength of 750900 MPaIn order to implement the strength increase factors dened inTable 2 at any Gauss point in a MDOF numerical model, the strainstate at needs to be transformed into an equivalent scalar value, andthen the equivalent strain rate can be calculated. The most commonstrain transformation is the incremental effective plastic strain used inthe theory of plasticity, and it can be dened either by thework harden-ing method or the strain hardening method. Although the effectiveplastic strain does provide a convenient transformation to a scalarquantity, both denitions result in a non-negative value. As a result, itis not a useful parameter for calculation of the equivalent strain-rate-dependent value, which can have a reversing sign. Therefore, in thecurrent study, the volumetric strain is used to transform the strainstate at any Gauss point to a scalar value. Since the steel materials arerelatively thin plates, only the in-plane strain components have beenconsidered in the equivalent volumetric strain calculation. The equiva-lent strain rate is dened by dividing the change in the equivalent volu-metric strain at each Gauss point by the time increment, and the strainrate effect is incorporated into the analysis based on the selected coef-cients in Table 2. Linear interpolation is used for any strain rate that fallsbetween the values in the table.

4.2. Damage model

As demonstrated in Table 1, the design guidelines dene thestructural performance in terms of support rotation and ductility ratio.The former and latter tend to limit the maximum deformation anddegree of nonlinearity (approximate measure of component uniaxialplastic strain), respectively, at the location of the component'smaximum demand. However, they do not capture all the parametersthat affect the blast damage in the component. For instance, when abiaxial stress condition exists or when the shear stress level is high atany point in the material, the steel could be prone to localized damageor failure other than strain softening under a large uniaxial strainvalue. Hence, the model must capture all the parameters that affectthe blast design and are not considered in the response limit values pro-posed in the design guides.

When the material subjected to the blast wave is damaged in theform of a breach, there is a potential for leakage pressure to enter thebuilding, especially for far-eld explosions with longer durations. Assuch, the model needs to capture the material damage properly, andidentify any blast load levels that cause material failure, even if theblast pressure itself induces acceptable ductility or support rotationresponse.

Moghimi andDriver [14] showed that global damage indices, such asthe change in dynamic properties of the system obtained from a modelincorporating only material plasticity, are not necessarily competenttools for accurately estimating the extent of damage in the systemwhen subjected to blast loads. Also, a simple fracture model based onmaximum strain criteria is not an accurate method, since the fracturestrain would be constant for all stress states. As such, the comprehen-sive damage models being used for the current numerical study,which take into account the stress state in the fracture model, arenecessary for developing accurate iso-response curves.

4.2.1. Damage initiation modelsHooputra et al. [18] suggested that metal sheets may fail due to one

or a combination of the different potential failure mechanisms, includ-ing ductile fracture due to void nucleation, growth, and coalescence,shear failure due to shear band localization, and instability due to local-ized necking. The study was carried out for specic types of aluminumalloy with yield strengths and strain hardening properties comparablewith structural steel. The material damage models were validated byquasi-static three-point-bending tests and quasi-static and dynamicaxial compression tests on the double-chamber extrusions. The pre-dominant fracture modes in all tests were found to be shear and ductile

failure, while instability failure did not govern due to the loading

50 H. Moghimi, R.G. Driver / Journal of Constructional Steel Research 106 (2015) 4456conditions. It was shown that for both quasi-static and dynamic load-ings, where the ratio of minor to major principal strain rate is largerthan about +0.35, the instability damage mode does not govern. For aSPSWsystemundermedium- to far-eld blast loads, the principal strainrate ratio is generally larger than 0.5, and as such, ductile and sheardamage are the governing damagemodes and are considered in the nu-merical model. When thematerial properties for structural steel are notavailable, the suggested properties for the damage models from theoriginal study [18] are used. Conservatively, the effect of strain rate oneffective plastic strain at the onset of damage is ignored for both failuremodels, since test data are not available for structural steel.

The ductile damage criterion [18] takes into account the effect of thehydrostatic stress condition on material damage by introducing a stresstriaxiality parameter, =m/*, wherem is themean stress and* isthe effective stress, as dened earlier. For a given temperature andstrain rate, the effective plastic strain at the verge of damage, pl0, canbe dened as a monotonically decreasing function of the stress triaxial-ity parameter as pl0 = d0 exp (c ), where d0 and c are scalar anddirectionally-dependent material parameters. Assuming homogeneousmaterial properties for steel, the parameter c becomes scalar and thevalue suggested by Hooputra et al. [18] is used (c = 5.4). Substitutingthe uniaxial coupon tension test ( = 1/3) into the equation for pl0gives d0=6.05 u, where u is the uniaxial plastic strain of the materialwhere the failure is initiated. In this study, the plastic ultimate strain ofthe materials from the tension coupon tests is used as u. As such, theeffective plastic strain at the onset of ductile damage would be:

pl0 6:05 u exp 5:4 : 7

The shear damage criterion [18] takes into account the effect of shearstress on the material failure by introducing the shear stress ratio pa-rameter, = (1 s )/, where s is an empirical material parameterand= max/* is the ratio of maximum shear stress to effective stress.In this study s=0.3 is selected, as proposed byHooputra et al. [18]. Theeffective plastic strain at the onset of damage, pl0, for a given tempera-ture and strain rate can be dened as a monotonically increasing func-tion of shear stress ratio as pl0 = d0 exp (f), where d0 and f arescalar material parameters. For the latter, the value suggested byHooputra et al. [18] is used (f=4.04). Substituting the uniaxial tensioncoupon test (= 1.8) into the equation for pl0 gives d0 = u/1439. Assuch, the effective plastic strain at the onset of shear damage is:

pl0 u=1439 exp 4:04 : 8

4.2.2. Damage evolutionWhen a material point is under a loading condition where its strain

tensor expands, the stress state eventually reaches the plastic limit.From this instance forward, the plasticity model takes over thematerialbehavior at that point and it denes any material softening and strainhardening up to the fracture strain. However, at the time the effectiveplastic strain in the material reaches the verge of damage, pl0, damagein the material is initiated and the material point enters the damageevolution phase. In this phase, the plasticity model cannot accuratelyrepresent the material behavior since it may introduce a strong meshdependency because of strain localization. As such, a damage evolutionlaw is added to thematerial behavior, which applies progressive degra-dation in material stiffness leading to complete material failure at theplastic strain equal to the effective plastic strain at failure, plf. A linearinterpolation is used in the current study for the material stiffnessdegradation evolution from the plastic strain at the onset of fracture tothe effective plastic strain at failure.

For both damage models, the same damage evolution law is used. Tomake the material response mesh-independent, ABAQUS formulates thedamage evolution law based on stressdisplacement (instead of stress

strain) response by introducing either fracture energy dissipation orthe effective plastic displacement at failure, uplf. The latter parameterwhich is dened with the evolution equation of dupl = L dpl, wherepl and L are the plastic strain at thematerial point and the characteristiclength of the element, respectivelyis used in the current study. A frac-ture energy-based approach is implemented to rationally estimate theeffective plastic displacement at failure. Based on classical fracture me-chanics, the strain energy release rate (fracture energy per unit area)for a crack in the rst mode of opening in the plain stress condition isequal to GI = KI2/E, where E is the modulus of elasticity and KI is thestress intensity factor for the rst mode of opening. For an innite platewith a crack with a length of 2a, the stress intensity factor is equal toKI=(a), where is the stress that initiated the crack. In the currentproblem, the applied stress is selected equal to pl0, which is the yieldstress corresponding to the effective plastic strain at the verge of damage.On the other hand, the fracture energy dissipation can also be denedbased on the effective stress and displacement as GI = pl0 uplf/2. Bothdenitions for the fracture energy dissipation result in the followingequation for the effective plastic displacement at failure:

upl f 2pl0 a =E: 9

Assuming the average crack length, yield stress at the onset of fail-ure, and modulus of elasticity for structural steel are equal to 20 mm,350 MPa, and 200 GPa, respectively, the effective plastic displacementat failure from Eq. (9) would be equal to uplf = 0.1 mm.

Although the damage evolution is formulated in terms of the effec-tive plastic displacement (instead of strain), themethod is still mesh de-pendent. The analysis showed that the model provides acceptableresults for global and member behavior, while it may not necessarilypredict the local material behavior at a small discontinuity accurately,since plastic and damage response sensitivity to the mesh renementexists.

Thematerialmodels and laws described above are incorporated intoABAQUS by means of dening appropriate eld variables and user sub-routines. The damage models are not compared with the cyclic test re-sults of Driver et al. [16], since most of the reported material failureswere due to low cycle fatigue rather than ductile and shear damage. Atypical result for a large out-of-plane blast load is shown in Fig. 3. Thepressure applied to the front wall is 1.66 MPa, which is a very severeblast pressure. The pressure applied to the top beam in the downwarddirection has a resultant equal to 1/2.5 times the resultant of thepressure applied to the inll plate, where the factor 2.5 is the reectioncoefcient and is described in the next section. The numerical modelshows the same failure pattern under out-of-plane blast overpressureas was reported by Warn and Bruneau [13], where the inll plateunzipped along three weld lines.

5. Numerical model of SPSW systems

5.1. Blast effects on SPSWs

Positive-phase shock-wave-type blast loads are used in all analysesof this study; the negative phase is ignored as it contributes little tothe overall dynamic response. Fig. 4(a) shows a typical shock load andits linearized triangular step-type load. P0 is themaximumpeak incidentor reected blast overpressure, which can be represented as a pressureor force. td is the positive-phase duration, or the duration of the linear-ized triangular step-type load. In this study, the linearized shock loadis used, where P0 and td are variables in developing PI curves. Thearea under the P0-td triangle is the intensity (I) of the blast load. Theblast loads on the wall and the roof are assumed to be in phase sincethe system is a single wall. Design blast loads (pressure and duration)for petrochemical facilities can be found in the literature [2,4] and aresummarized byMoghimi and Driver [14]. In most blast-resistant design

cases, which are close to the impulsive asymptote, the maximum

duration of the blast loads is less than one-quarter of therst natural pe-riod of the structure, and the pulse load shape has a negligible effect onthe dynamic response.

SPSW systems in a protective structure could be under in-plane orout-of-plane blast load effects. An idealized, square-plan protectivestructurerepresenting a building such as a small control houseconsisting of four SPSWs under reected overpressure on the frontwall and incident overpressure on the roof is shown in Fig. 4(b).These blast waves are separated into out-of-plane and roof overpres-

Fig. 3. Effective plastic strain and damage distribution

51H. Moghimi, R.G. Driver / Journal of Constructional Steel Research 106 (2015) 4456sures on the front wall, as shown in Fig. 4(c), and in-plane and roofoverpressures on the side walls, as shown in Fig. 4(d). The blast re-sponse of these two wallsthe side and front wallare investigatedindividually.

The side wall is analyzed under uniform in-plane reected blastoverpressure acting on the left column (in the left-to-right or+X direc-tion) and uniform roof incident blast overpressure acting on the topFig. 4. (a) Shock wave blast load, (b) Protective structure consisting of four SPSWs underblast loading, (c) Front wall under out-of-plane and roof blast loads, (d) Side wall underin-plane and roof blast loads.beam (in the downward or Y direction), as indicated in Fig. 4(d).The effects of the front and rear walls are replaced by a roller supportat the center of each frame connection that prevents movement in theZ-direction. The resultant of the blast overpressure acting on eachstructural member (left column and beam) depends on the blast-loadtributary area for the member. To make the results independent of thetributary area, the resultant reected blast pressure applied to the leftcolumn is dened as amultiplier of the yield strength of thewall, Vy, de-rived from a bilinear curve taken from the pushover analysis result withthe same spatial load distribution as the blast load, and explained inSection 5.3.1. The resultant blast load acting on the top beam is equalto the same multiplier times Vy/Cr, where Cr is the reection coefcientpertaining to the front wall, which is a function of peak overpressureand the angle of incidence. For the range of parameters used in petro-chemical facilities, Cr can be selected as 2.5 [2], implying a blast wavethat is perpendicular to the front wall. The blast effect is localized tothe vicinity of the reected blast pressure andmaximizes the local duc-tility demand in the left column.

The frontwall is analyzed under uniformout-of-plane reected blastoverpressure acting on the inll plate (in theZ direction) and uniformroof incident blast overpressure acting on the top beam (in the down-ward orY direction), as indicated in Fig. 4(c). The out-of-plane trans-lationalmovements (in the Z-direction) of the top and bottomanges ofthe beam at the connections are restrained. Similar to the side wall, theresultant blast pressure on the inll plate and top beam are different

in a front wall from large out-of-plane blast load.multipliers of Vyo and Vyo/Cr, respectively, where Vyo is dened inSection 5.3.2 for the out-of-plane response of the wall and Cr is selectedas the same value as for the side wall (2.5).

5.2. Selected SPSW systems

The competency of SPSW systems under blast loads is investigatedby developing PI curves for different walls and different responselimits and comparing them with the results of relevant design exam-ples. The selected base model is the rst story of a multi-story SPSWspecimen tested by Driver et al. [16] and described in Section 4. Sincemost industrial protective structures are one-story buildings for designand safety requirements, only the rst story of the specimen is consid-ered in this study. However, to allow the development of full tensileyielding of the inll plate, the deep top beam of the actual specimenwas used for the analyzed system. As shown in Fig. 5(a), the columnsare W310 118 sections spaced at 3.05 m center-to-center. The topbeam is a W530 82 section and the total story height is 2.15 m. Mo-ment connections are used at the beam-to-column joints, and the inllplate thickness is 4.54 mm.

52 H. Moghimi, R.G. Driver / Journal of Constructional Steel Research 106 (2015) 4456The above-described system represents a half-scale wall for an in-dustrial protective building. To study the size effect on the response,a full-scale wall with the same conguration but all the dimensions(including beam and column cross-sectional depth and elementwidth and thickness) are doubled. The story height is therefore4.30 m and the columns are spaced at 6.1 m center-to-center.These two walls are the subject of the blast-resistance study underin-plane and out-of-plane blast loads. The share of the roof's deadload on one wall is simulated by adding total masses of 3240 kgand 12 960 kg, for the half- and full-scale walls, respectively. Themass is added by means of point masses distributed to the topbeam at the nodes on the beam-to-column connections and alongthe beam ange-to-web interface, effectively creating a uniformmass distribution over the beam length and a more concentredmass at the beam-to-column connections.

5.3. Pushover responses of SPSW systems

The pushover analysis results are needed for the normalization pro-cess to transform eachwall system into an ESDOF system. Also, differentmultipliers of the yield strength of each wall are used as the blast loadintensity applied to the system. As such, the yield strength and yielddisplacement of the wall systems are needed, and are dened in the fol-lowing sections. This process needs to be performed carefully for eachresponse type, since it will form the basis of the transformation process.

Fig. 5. Selected half-scale SPSW system, (a) Wall elevation and in-plane pushover load-ings, (b) Pushover analysis results and comparison with test.5.3.1. In-plane response (side wall)The in-plane deection at the roof level is selected as the

displacement in the pushover curve and the generalized displace-ment for the ESDOF system. The dot-dashed pushover curve inFig. 5(b) shows the in-plane pushover analysis of the systemunder monotonically increasing displacement at the roof level.The curve demonstrates a similar result to that of the base shearversus rst-story displacement response of the four-story test spec-imen. However, in order for the results of the pushover curve to beapplicable to a SDOF system, the lateral pushover force should havethe same spatial distribution as the applied blast load. The solidcurve in Fig. 5(b) shows the result of a pushover analysis under auniform in-plane pressure applied to the left column. The pushoverload distribution is similar to the in-plane blast load and is identi-ed as Modied pushover analysis in Fig. 5(a). Because of the dis-tributed applied load, the modied pushover curve has a largeryield strength and displacement.

The yield shear strength and displacement of the side wall can befound from the equivalent bilinear pushover curve representing thereal nonlinear curve with the same energy absorption capacity. Bilinearrepresentation of the pushover curve results in yield values equal toVy = 3300 kN and y = 7 mm. The modied pushover analysis curveof the full-scale wall shows the same behavior as the half-scale wall inFig. 5(b), but with the yield values equal to Vy = 13 200 kN and y =14 mm.

5.3.2. Out-of-plane response (front wall)The out-of-plane pushover analysis of the SPSW demonstrates a

distinctly different response. The out-of-plane deformation at thecenter of the inll plate is selected as the displacement in the push-over curve and the generalized displacement for the ESDOF system.Since the inll plate is the main element that contributes to theout-of-plane resistance of the front wall, it is a simple system andthe pushover curve can be evaluated by an analytical approach in ad-dition to the numerical method. Two analytical approachesnamely,the energy balance and equilibrium methodsare used to derive thepushover curve. The out-of-plane exural resistance of the inllplate is ignored in the total resistance in both analytical approaches,since membrane action dominates the response. Fig. 6(a) demon-strates a typical SPSW system with an inll plate thickness of wand clear length and width of inll plate equal to Lc and hc, respec-tively. The corresponding yield lines for the inll plate subjected tothe out-of-plane blast pressure with intensity p are shown in the g-ure. For a given wall system, the deformed shape of the inll platedepends on the panel aspect ratio and the dynamic load rate and in-tensity. For high-speed loading, the deformed shape is close to a lin-ear deformation, as depicted in Fig. 6(a), while for quasi-staticloading the deformed shape becomes curved.

The external work in the energy balance method is equal toWE = p hc (3Lc 2x) / 6. The internal strain energy can be calculatedby dividing the inll plate into two triangleswith height x and two trap-ezoids, as shown in Fig. 6(a). The strain energy is equal to the axialtension in each element multiplied by its axial deformation, orWI = w hc [x L + (Lc x) h], where h = [(2 / hc)2 + 1] 1and L = [( / x)2 + 1] 1 are axial strains in the vertical and hor-izontal directions, respectively. The parameter x can be evaluated byequating the external work to the internal energy and solving for thepressure, p. Setting the rst derivative of the pressure equation withrespect to x equal to zero gives the maximum pressure applied to thesystem. However, the equation with respect to xwould be nonlinearand does not have a closed-form solution. As such, a simplifying as-sumption is made for x based on the numerical response of theSPSW under blast load at different loading rates. The numerical re-sults show that at a fast blast loading rate (impulsive asymptote),the deformed shape of the inll plate is such that the plate rotation

along the vertical boundary L = or x = hc / 2 (Fig. 6(a)), which

as was the case for the side wall. This is attributed to the fact that withan increase in the applied force, the plate experiences increased out-

53H. Moghimi, R.G. Driver / Journal of Constructional Steel Research 106 (2015) 4456results in L = h. The assumption results in the following equationfor the resultant force of the applied pressure to the inll plate:

pLchc 6 wLc2hc= 3Lchc 4=hc2 1=2h i

1=

10

where is the tensile stress in the inll plate and is equal to the elas-tic stress for strains less than the yield strain and the yield stress forgreater strain values.

The resultant force applied to the inll plate can also be evaluatedbased on equilibrium and the same assumed deformed shape.Neglecting the exural resistance of the inll plate, at every stage ofloading the horizontal component of the membrane action resists theapplied lateral loads. As such, the summation of the horizontal compo-nents of the membrane forces fIh and fIL in Fig. 6(a), which are appliedto the horizontal and vertical boundaries, respectively, is equal to theresultant force of the applied pressure:

phcLc 2 w Lc hc sin 11

where tan ()= 2/hc. Fig. 6(b) shows the pushover curves for the half-scale wall depicted in Fig. 5(a) from the numerical model and the twoanalytical methods (Eqs. (10) and (11)). Also, the result of the methodproposed by Warn and Bruneau [13], which assumes a second-orderpolynomial for the inll plate deformation in both directions, is shown.

The results show that in the early stage of the pushover curve, the re-sponse is nonlinear. Referring to the curve in Fig. 6(b) obtained from thenumerical model, the at (unloaded) inll plate subjected to out-of-

Fig. 6. (a) Inll plate subjected to out-of-plane blast load and its associated yield lines,(b) Pushover curve for out-of-plane resistance of inll plates versus displacement atcenter.of-plane deformation, thus increasing the out-of-plane component ofthe inll plate membrane force. This phenomenon results in a fairly lin-ear pushover curve, even beyond the chord rotation of 12 for the inllplate connection to the boundary frames, which is considered to be thelargest acceptable out-of-plane deformation (or rotation) for the platesaccording to all blast design guides cited in Table 1. For comparison, therotation limits of 3, 6, and 12 are marked in Fig. 6(b).

The results in Fig. 6(b) show that the assumption of a parabolicshape for the inll plate overestimates the resistance, while the energybalance method underestimates the resistance. The equilibrium meth-od, with an assumed linear deformed shape, estimates the resistanceof the inll plate closest to the numerical result. Their actual differenceis even smaller in reality for two reasons. First, the load is applied slowlyin the numerical model, which forces the inll plate to deform into acurved shape. As explained previously, the inll plate deformationwould be nearly linear for a fast rate of loading (impulsive asymptote),which is the subject of the current study. In such a case, the numericalresult would be even closer to the equilibrium method result. Second,the numerical analysis was carried out using a displacement-controlled method, and the applied displacement was in the horizontaldirection. As a result, with increasing out-of-plane deformation of theinll plate, the component of the applied force that is parallel to themembrane force increases. However, the blast pressure is always nor-mal to the inll plate, which reduces the resistance of the inll plateslightly at each displacement level.

The analytical (equilibrium) approach estimates the out-of-planedisplacement associated with achieving the yield strain in the inllplate to be equal to 48 mm. The numerical model estimates the out-of-plane yield displacement and its corresponding strength to beequal to yo = 58 mm and Vyo = 986 kN (see Fig. 6(b)), respectively,where the subscript o represents the out-of-plane response. Thispoint is associatedwith the change in the curvature sign of the pushovercurve, which corresponds to a fully yielded inll plate in the verticaldirection. The pushover analysis curve of the full-scale wall shows thesame behavior as the half-scale wall, but with an out-of-plane yield dis-placement and strength equal to yo = 117 mm and Vyo = 4030 kN,respectively.

6. Results of blast design study

6.1. Response limits

The goal of this study is to evaluate the blast performance of SPSWsystems by developing iso-response curves. Any PI curve applies onlyto one single response limit in the system. In blast design, the membersof structures are allowed to yield to achieve an economical design. Thedesign guidelines specify the structural performance of the ductile ele-ments based on maximum deformation limits instead of strengthlevel, to provide controlled and ductile yielding. In an approximateway, these limits dene the strain energy absorption capacity andplane load has a very small resistance because of its low exuralstiffness, and consequently it deforms rapidly under the load. As theinll plate deforms further in the out-of-plane direction, the appliedload is increasingly carried by membrane action. This effect increasesthe stiffness of the system considerably, as reected by the upwardcurved shape of the pushover diagram. (In order to capture this effectin the numerical model, geometric non-linearities must be taken intoaccount.) The eventual decrease in stiffness occurs due to materialyielding.

Although at large out-of-plane deformations the inll plate is fullyyielded and the membrane forces remain fairly constant, the pushovercurves do not indicate a well-dened yield strength and yield plateau,amount of acceptable damage in the component and thereby prevent

6.2. PI curves

6.2.1. Iso-response curvesThe iso-response curves are developed for the four different walls

and the response limits specied in the previous section using themodel described earlier and the trail-and error approach. In this study,all analyses are done with the explicit method, originally developed toanalyze nonlinear high-speed dynamic systems. Conservatively, theeffect of damping in the analysis is not considered. However, this effectis negligible since the peak dynamic responses are studied. For theSPSW systems under consideration, the shear damage is usually a littlemore critical than ductile damage in both blast loading directions.

The PI curves for the half- and full-scale side walls are demonstrat-ed in Fig. 7(a). The response limits are local and chord (global) rotations(-L and -G, respectively) of the beam-to-left-column connection of 1.As expected, the full-scale wall is considerably stronger than the half-scale wall. The chord support rotation results in a larger rotation thanthe actual local rotation at the joint, and it is a more conservative re-sponse limit. In other words, for a given rotation limit, the local rotationshows a larger applied pressure and impulse than the chord rotation.

Some extensions of the impulsive and quasi-static asymptotes forthe local rotation are shown by dashed lines for both the half- andfull-scalewalls. Although they represent 1 connection rotations, ductileand/or shear failure (especially of the left column and inll plate adja-cent to the left column base) occurs in the system in these branches.Since the damage is sustained by the system, the dashed lines are notacceptable regions for design. For instance, themaximumblast force re-sultant that can be applied to the full-scale wall is 7.05Vy = 93 060 kN.Any blast load with a larger resultant, even with a smaller duration,would produce a 1 connection rotation and cause failure in the system.

54 H. Moghimi, R.G. Driver / Journal of Constructional Steel Research 106 (2015) 4456component failure. The safety factor in design is also included in the de-formation response limits.

As pointed out earlier in reference to Table 1, the deformation per-formance criteria are usually dened in terms of ductility ratio and sup-port rotation for individual members. Both response parameters can becalculated from themaximum component deformation, which is deter-mined from the nonlinear dynamic numerical model in this study. Theresponse criteria are applicable to ductile members, and when brittlefailures are prevented ductile exural response governs the overallcomponent behavior. The equivalent static method can be used to pre-vent brittle failure in strength-controlled actions, such as axial compres-sion, shear, and reaction forces at the connections. In this method, theinternal force demands should be less than the lower-bound strengthsfor the corresponding actions.

The yield displacement is the effective displacement atwhich plasticstrain begins in the same location as themaximum deformation occurs.The yield displacement of the system and its corresponding yieldstrength (y and Vy) can be extracted from the static pushover resis-tancedisplacement graph of the system, where the pushover load hasthe same spatial distribution as the applied blast load. The method re-sults in an equivalent yield displacement that produces the equiva-lent ductility ratio. If the yield displacement is selected as thedisplacement at the beginning of the yielding process or the dis-placement at the mechanism condition, the resulting ductility ratiowould be respectively greater and less than the equivalent ductilityratio. The support rotation can be calculated from the chord rotationof the member's overall (or global) exural response rather than theactual local rotation at a point in the connection. The tangent of thesupport chord rotation is dened as the ratio of themaximum deec-tion in the member to the shortest distance from support to the loca-tion of maximum deection.

The support rotation controls the maximum deformation (whichusually occurs at midspan) of the member. The ductility ratio limitsthe extent of plastic response in the maximum deformation locationof the component (which should be the same location as where the ini-tial yielding occurs). However, a ductile member may develop mem-brane action at a large ductility ratio. The axial tensile action maycause damage/failure in the connections before damage/failure at themidspan occurs. In such a case, the support rotation can limit theamount of membrane tensile force to an acceptable level that preventsthe connection failure. Hence, it would be a better damage index for aductile member.

Support rotation limits for the low response range of the structuralsteel system have been selected to develop the iso-response curves inthis study. The rotation of the beam-to-column connection adjacent tothe column under blast load (point E in Fig. 4(d)) is limited to 1 forthe side-wall study. Both local and global rotations are considered andtheir results are compared. The local rotation is the amount of actual ro-tation at point E. The tangent of the global (overall or chord) rotation iscalculated by dividing the maximum translation at the middle of thebeamor columnby the distance between the connection andmaximumdeformation point. Since the reected blast pressure is applied to theleft column in this study, the maximum deformation occurs at themiddle of the left column, and the tangent of the global rotation atpoint E or F is calculated by dividing the Ux deformation of point D inFig. 4(d) by the half-column height.

The rotation of the inll plate connection at the base or top beam atthemiddle of the bay (points B and C in Fig. 4(c)) is limited to 3 for thefront wall study. The tangent of the inll plate rotation is calculated bydividing the out-of-plane deformation of the inll plate mid-point(point A in Fig. 4(c)) by the half-span height. For comparison and tocheck the extent of yielding in the inll plate at the support rotationof 3, the iso-response curves for a ductility ratio of 1.0 are calculated.The yield displacement is selected from the pushover curve, and theiso-response curve shows the pressure and impulse combinations that

cause the inll plate to be fully yielded. Fig. 7. PI diagrams, (a) Side wall for 1 rotation, (b) Front wall.

Therefore, any combination of P and duration at the impulsive asymp-tote that produces the same impulse may not be acceptable in design,especially for the blast load with large incident overpressure. Thesame concept is valid for the quasi-static loading realm. The minimumblast load resultant that can be applied to the system and reach themonitored response limit is 2.6Vy = 34 320 kN. A smaller blast loadwith duration large enough to produce a 1 connection rotation causesfailure in the system.

Fig. 7(b) shows the iso-response curves for both frontwalls and bothresponse limits of 3 connection rotation and a ductility ratio of unity.The chord rotation curve also limits the maximum out-of-plane defor-mation of the inll plate, which is one of the limit criteria in the frontwall design. The gure shows that the 3 connection rotation imposesa more conservative response limit than the ductility ratio of unity.

The efciency of SPSW systems in petrochemical facilities is in-vestigated by checking appropriate design examples for both half-and full-scale walls against the iso-response curves. The actual walldimensions are used for the front wall design. However, an assump-tion for the tributary width is necessary for each side wall design.Widths of 7 m and 14 m are assumed for the half- and full-scalewalls, respectively, constituting a relatively severe demand scenario.The blast-pressure design is based on the ASCE guideline [2], whichspecies a shock wave with 70 kPa side-on overpressurewhich isscaled up by a reection factor of 2.5 to represent the reected pres-sure of 175 kPaand a 20 ms duration. Since the design points all fallbelow the associated PI curves in Figs. 7, the selected SPSWs consti-tute a competent force resisting system for accidental blast loadingin petrochemical facilities, even for the low response limit and out-of-plane blast load.

55H. Moghimi, R.G. Driver / Journal of Constructional Steel Research 106 (2015) 4456Fig. 8. Nondimensional PI diagrams, (a) Side wall for 1 rotation, (b) Front wall.6.2.2. Nondimensional curvesUsing Eqs. (2) and (6), the iso-response curves for the side and front

walls are normalized and shown in Figs. 8(a) and (b), respectively. Thesignicant DOF of each wall, which is the DOF of the ESDOF system, hasbeen selected as the roof displacement for the side wall and the out-of-plane displacement of the center of the inll plate for the frontwall. Theshape function for the ESDOF system is selected as the normalized de-formed shape of the system under static application of the blast load.Unlike the fundamental mode shape, thismethodworks for both elasticand plastic responses and provides better results for design forcecalculations.

Nevertheless, the ESDOF system is approximate, since the deformedshape of any system not only depends on the spatial load distributionbut also on the rate of the applied load and interaction of the loadwith the structural response. Considering a given iso-response curve,the deformed shapes of the system in the quasi-static and impulsiveloading regimes are different. Moreover, in the impulsive loadingasymptote, local deformation exists in the system because of the fastapplication of the blast load which cannot be captured with the staticdeformed shape of the system. However, the approximations have anegligible effect on the dimensionless curve results as long as thesame shape function used to produce the dimensionless curves is alsoused to employ them for design purposes.

Figs. 8(a) and (b) show that the normalization method brings eachpair of curves close together. This can be an effective method for devel-oping unied iso-response curves for different SPSWs. While in theorythe normalized PI curves can be applied to any SPSW with the sametype of system as the one used to develop the curves, the best accuracywill be achieved for walls with comparable values for the parametersthat most inuence the behavior, such as similar panel aspect ratios.

6.3. Charge weightstandoff distance diagram

The information in an iso-response curve can also be represented inthe form of combinations of explosive charge weight (W) and distancefrom the charge (standoff distance, R) that cause the response limit tobe reached. The WR curve denes the system susceptibility to airexplosions and shows all combinations of energy release amounts andstandoff distances that cause the same maximum response in thesystem. Under sea-level ambient conditions, the charge weight andstandoff distance are uniquely determined based on the type ofexplosion. In this study, the positive phase of the shock wave param-eters for a hemispherical TNT surface burst at sea level (CONWEP[19]) is selected, since it is similar to potential accidental industrialexplosions.

Each point on a PI diagram associates the reected blast pressure(Pr) to the impulse (Ir) that will cause the selected response limit tobe reached. From the reected blast pressure, the corresponding scaleddistance, Z = R/W1/3, is determined from the selected blast chart.Having the scaled distance, the normalized reected blast pressure,Ir/W1/3, is obtained. The charge weight is evaluated by substituting theblast impulse from the PI curve into the normalized reected blastpressure. The standoff distance is then determined by substituting thecharge weight into the scaled distance equation.

The conversion of the PI diagrams in Fig. 7 into WR curves isshown in Fig. 9. Logarithmic axes are selected since this representationofWR curves tends to be close to a straight line with a positive slope.WR curves represent iso-response curves similar to PI diagrams.Any point above the curve implies the response limit has beenexceeded, while a point below the curve indicates a response lowerthan the limit.

7. Conclusion

It has been proven through past research that SPSWsmake superior

lateral force resisting systems for seismic applications. Despite the

response characteristics need to be considered. Breaching of themateri-al under close-proximity blast events with large, highly-localizedpressures has not been considered in this study and could be includedin future work. Low cycle fatigue failure has not been considered inthe analysis and it may have some effect in the overall response of theSPSW system under far-eld accidental blast load. The walls studied inthis research were consistent with those designed for seismic loading;to optimize the system and improve its reliability, detailing developedexplicitly for blast-resistant applications is needed.

56 H. Moghimi, R.G. Driver / Journal of Constructional Steel Research 106 (2015) 4456inherent slenderness of the steel members, this study shows that theSPSW system has potential to be an effective protective structure in in-dustrial plants. The side wall subjected to in-plane blast load is a strongand reliable system, and the front wall subjected to out-of-plane blastload can be sized to provide acceptable design for industrial plantapplications.

The proposed nondimensional PI diagramprovides an efcient toolfor preliminary design of SPSW systems to resist accidental blast loads.The study shows that to develop reliable iso-response curves, appropri-ate damage and failure criteria should be considered in the numericalstudy of the wall system. This is especially important in structuralsteel members, since their slenderness makes them vulnerable todamage under heavy blast overpressure, even with a very smallduration.

Additional research is required to study the effects of various param-eters not considered in the limited study presented in this paper. Wallswith different geometries, mechanical properties, and dynamic

Engineering, University of Alberta; 2002.

Fig. 9. Charge weightStandoff diagram, (a) Side wall for 1 rotation, (b) Front wall.[18] Hooputra H, Gese H, Dell H, Werner H. A comprehensive failure model for crashwor-thiness simulation of aluminium extrusions. Int J Crashworthiness 2004;9(5):44964.

[19] CONWEP. Conventional weapons effects. Vicksburg, MS: U.S. Army Corps of Engi-neers, Waterways Experiment Station; 1992.Acknowledgements

Funding for this research was provided by the Natural Sciences andEngineering Research Council and the Steel Structures EducationFoundation.

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Performance assessment of steel plate shear walls under accidental blast loads1. Introduction2. Literature review2.1. Performance criteria2.2. PI curves2.3. SPSW systems

3. PI diagrams3.1. PI diagram versus response spectrum curve3.2. Trial-and-error approach3.3. Nondimensional PI diagram3.3.1. Nondimensional load/pressure3.3.2. Nondimensional impulse

4. Material model for blast analysis of SPSWs4.1. Constitutive material model4.1.1. Hardening rules4.1.2. Rate dependent yield

4.2. Damage model4.2.1. Damage initiation models4.2.2. Damage evolution

5. Numerical model of SPSW systems5.1. Blast effects on SPSWs5.2. Selected SPSW systems5.3. Pushover responses of SPSW systems5.3.1. In-plane response (side wall)5.3.2. Out-of-plane response (front wall)

6. Results of blast design study6.1. Response limits6.2. PI curves6.2.1. Iso-response curves6.2.2. Nondimensional curves

6.3. Charge weightstandoff distance diagram

7. ConclusionAcknowledgementsReferences