Transverse blast loading of hollow beams with square cross-sections

Thin-Walled Structures 62 (2013) 169–178

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Transverse blast loading of hollow beams with square cross-sections D. Karagiozova a, T.X. Yu b,c, G. Lu b,n a b c

Institute of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bonchev Street, Block 4, Sofia 1113, Bulgaria School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore 639798, Singapore Department of Engineering Mechanics, Zhejiang University, Hangzhou 310058, China

a r t i c l e i n f o

abstract

Article history: Received 24 July 2012 Received in revised form 12 September 2012 Accepted 12 September 2012 Available online 12 October 2012

A model of deformation of a metal hollow section beam under a uniform blast loading is developed in order to reveal the characteristic features of deformation and energy absorption of hollow section beams under such loading. It is established that as a typical structural component a hollow section distinguishes itself from its solid counterpart with two characteristic features of the response. First, a considerably larger kinetic energy is generated in the hollow section beam as the impulsive load is imparted on the upper flange of the beam having a significantly lower mass than the member. Second, a considerable proportion of the blast energy can be absorbed by the local collapse of the section. A twophase analytical model is proposed. In the first phase, the local collapse of the thin-walled cross-section is determined by using an upper bound approach; and in the subsequent second phase, the global bending of the beam with the distorted section is analyzed by taking into account the effect of axial force. It is demonstrated that mass distribution in the hollow section is an important factor in determining the energy partitioning between the local deformation phase and global bending of the hollow beam. Reasonable agreement is obtained with the experimental data published in the literature [Jama HH, Nurick GN, Bambach MR, Grzebieta RH, Zhao XL, Steel square hollow sections subjected to transverse blast loads, Thin-Walled Structures 2012;53:109–122]. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Hollow beam section Blast load Energy absorption Analytical model Rigid-plastic analysis

1. Introduction The analysis of various structural members subjected to blast loads are of an interest due to increased security demands in the urban environment as a result of the occurrence of accidental or intended explosions. Thin-walled members with various crosssections including hollow sections are used extensively in the construction, offshore and mining. The responses of these structural members to blast loading can include unacceptably large permanent deformations and even total failure. Although the numerical analysis can provide valuable information on the details of the response of structural members with more complex material properties, the analytical models, which retain the characteristic features of the structural response, can reveal important relationships between the structural parameters. Analytical models of the response of metal beams with solid sections subjected to transverse blast loads have become classical guidelines to analyze the influence of different factors on the behavior of these members. Due to the large plastic deformations involved often the elastic deformations are neglected and a rigid plastic material model is

n

Corresponding author. E-mail address: [email protected] (G. Lu).

0263-8231/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.tws.2012.09.004

employed. Depending on the geometry and material characteristics different factors may become important under intensive dynamic load. It has been recognized that similarly to the influence of the geometry changes on the behavior of statically loaded structural elements [1], the finite displacements play an important role for structural elements loaded dynamically [2]. Among the analytical studies on the axial membrane and bending response of rigid-plastic beams subjected to transverse impulsive loads the notable work include that by Symonds and Mentel [3] on pinned and clamped beams, Jones [4] on beams and plates, Symonds and Jones [5] on the combined effect of finite deflections and strain rate. These studies have shown the importance of retaining the axial membrane force in the yield condition and the beam response, particularly for beams with large length/depth ratios that respond with finite transverse deflections greater than the beam depth. The developed analytical models are predominantly focused on solid metal cross-sections and the investigations of impulsively loaded beams of other sections are limited to sandwich beams with different core configurations. As metal foams become available, sandwich beams and plates using metal foam as core have received much attention and their response to blast has been investigated extensively [6,7]. Sandwich beams with cellular materials/structures as core such as honeycomb, lattice, ‘‘Y’’ frame or corrugated plates have been analyzed. Blast

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response of circular and square sandwich panels with foam core has been studied [8,9] and experiments with sandwich shells were conducted [10]. Most recently, the behavior of a circular sandwich cylinder with internal blast loading has been investigated [11]. See Ref. [12] for a review of the recent studies on the response of sandwich structures and Ref. [13] on the impact dynamics and its applications. Despite some similarities in the response of hollow section beams and sandwich beams, the response of thin-walled beams to blast loading is less studied. The behavior of an impulsively loaded simply supported steel beams with a hollow section was studied by Wegener and Martin [14]. A semi-empirical analytical solution was derived with a partial use of a numerical analysis to determine the deformation modes of the beam. More recently, experimental work on steel hollow and steel concrete filled sections was reported by Bambach et al. [15] and on aluminium hollow section beams by Bambach [16]. An extensive experimental program on the blast impact of clamped hollow beams with square sections was carried out by Jama [17] and the major results from this study were published in [18]. In addition to the experimental studies reported in [16–18], a semi-empirical analysis gave bounded solutions for the observed transverse plastic deformation of hollow members using the assumption that the local collapse of the beam section and the global bending of the beam develop sequentially. This assumption is confirmed to a large extent by the experiments and numerical analysis of the hollow section in [19]. The aim of the present analysis is to develop a model, which adequately describes the deformation phases of a hollow section beam subjected to an impulsive loading, considering the temporal development of the deflections of the beam. A twophase analytical model is proposed in this paper. In the first phase, the local collapse of the thin-walled cross-section is determined by using an upper bound approach; and in the subsequent second phase, the global bending of the beam with the distorted section is analyzed by taking into account the effect of axial force. The model allows an estimation of the absorbed energy during the response. The details of the proposed model are presented in Sections 3 and 4 where the essential influence of the strain rate is taken into account. The energy partitioning between the local and global deformations depends on the load intensity and it is analyzed for different mass distributions of the hollow sections when retaining the total mass of the beam constant.

2. Blast loading of a hollow section beam An analysis is carried out of the response of an impulsively loaded metal hollow section beam with square or rectangular shape, which is an idealization of the behavior of hollow sections subjected to a uniform blast. Experimental results of blast loading on square steel hollow beam reported in the literature [17,18] are used to verify the proposed model. The structural members studied in [17,18] were approximately uniformly loaded using PE4 plastic explosive placed in one, two or three equally spaced strips on a polystyrene path on the upper flange (face) of the hollow section. The predictions of the proposed model are compared with the results from the experiments with two and three explosive strips as these loading conditions led to a more uniform loading. Therefore, it is anticipated that the load is uniformly distributed across the full width of the upper flange of a clamped hollow beam (Fig. 1(a)) and can be idealized as an impulsive loading due to the very short pulse duration resulting from the detonation of the explosive.

B

h

H

L

l ξ

W

Fig. 1. A hollow beam under blast loading; (a) geometry and loading configuration and (b) mechanism of global bending.

Based on the experimental observations [17,18] and the conclusion from the numerical simulations of the behavior of hollow beams reported in [19] it is anticipated that a phase of a local collapse of the cross-section precedes the global bending of the beam. For that reason, the model of the beam response is developed by assuming that there is no coupling between the two phases of deformation. As used in the experiments [17,18] hollow beams with section dimensions 35  35, 40  40 and 50  50 mm, all with wall thickness of 1.6 mm, are utilized for the model verification.

3. Local deformation phase 3.1. Mechanism of deformation Based on the experimentally observed pattern of deformation (Fig. A1(a), see Appendix) a model for the local collapse of the beam section is proposed to evaluate the energy absorbed during this deformation phase. The cross-section is characterized by width B, height H and wall thickness h. The construction of the geometrical profile of the deformed section due to the vertical displacement, u, is shown in Fig. A1(b). It is assumed that the section collapses due to the plastic bending of segments CD and MF with curvatures k1 and k2, respectively, and a stationary hinge at the corner F of the section. The elastic deformations of the beam are neglected and rigid, perfectly plastic property is assumed for the base material. The deformation energy per unit length of the hollow beam is equal to the plastic bending energy and is calculated as   Z Z EL ðuÞ=L ¼ 2M 0 bðuÞ þ Dk1 ðuÞds þ Dk2 ðuÞds ð1Þ CD

MF

where M0 ¼ s0h2/4 is the fully plastic bending moment per unit length of the section wall, and s0 is the material flow stress. L is the length of the beam. b(u) is the change of angle at plastic hinge F, which varies with displacement u. From the geometric relationships (see Appendix) the energy can be obtained as a function of the section dimensions B, H, h and the radii of segments CD and FM, R1 and R2. Since the energy is not uniquely defined, a minimum is sought with respect to R1 to obtain the values for the model parameters and this process leads to R1 ¼0 for all the displacements. However, due to the finite thickness of the wall and relatively large slenderness ratio B/h of the analyzed section it can be assumed that R1 should be larger than zero. Because of the very large curvature at the centre of the upper flange it is reasonable to assume that R1 Z 2h. A constant length of the rigid link DF (Fig. A1(b)) can be obtained from the

D. Karagiozova et al. / Thin-Walled Structures 62 (2013) 169–178

requirement that R1 ¼ 2h for the fully collapsed section. For example, LDF ¼ 0:35B for a cross-section 40  40  1.6 mm. 3.2. SDOF model The deformation model for the quasi-static local collapse is used to obtain the energy absorbed in the case of dynamic loading by assuming that the dynamic shape of deformation replicates the quasi-static shape shown in Fig. A1. The enhancement of the energy absorption is caused by the strain rate effect and is taken into account by the Cowper–Symonds equation

sd ¼ s0 ð1 þðe_ =DÞ1=p Þ

ð2Þ

which defines the increase of the dynamic flow stress sd due to the strain rate e_ ; D and p are material constants. It is assumed that the dynamic local deformation of the section can be represented by a generalized equivalent force, which produces the same deformation energy as that defined by Eq. (1). The blast loading is applied as an impulsive loading to the equivalent mass of the model. Since the plastic deformation is the only mechanism for energy dissipation, the plastic energy defined by Eq. (1) can be regarded as a total energy of the system under quasi-static deformation and the energy per unit length of the beam is expressed as EnL ¼ EL =L ¼ FðuÞu

ð3Þ

where u is the reduction in the height H of the cross-section measured at the highest point of the cross-section and F(u) is the equivalent force per unit length of the beam at the corresponding point. The collapse force, F, is defined as FðuÞ ¼ dEnL =du:

ð4Þ

The mass corresponding to length CDFM is involved in the deformation of the section and it varies with u. Therefore, the inertia contribution should be taken into account via an equivalent variable mass, which is a fraction of the mass m, e.g., mðuÞ ¼ mkm ðuÞ,

ð5Þ

where m ¼ rhB is the mass per unit length of the upper flange of the beam where the load is applied. However, the acceleration along the deformed shape is not uniform so that the correction coefficient for the equivalent mass is taken as an average constant from the expression _ _ km ðuÞ ¼ ðB=2 þ L FM ðuÞ=2Þ=ðB=2 þ L FM ðuÞÞ

ð6Þ

when the displacement u varies from zero to the displacement _ which corresponds to the full collapse of the cross-section; L FM is the arc length shown in Fig. A1(b). For the particular mechanism of deformation, km is found to be equal to 0.725. The equation of motion of the SDOF model is obtained with respect to transverse displacement u (see Fig. A1(b)) 2

mkm u€ þ Fðu=BÞs0 h =4B ¼ 0

with

m ¼ rhB,

u ruC

ð7Þ

where Fðu=BÞ is the non-dimensional equivalent force and uC is the value of the displacement u which corresponds to the fully collapsed section. In order to maintain the initial kinetic energy equal to that caused by the blast with total impulse I, the following initial conditions for Eq. (7) are used u¼0

and

1=2 _ uð0Þ ¼ I km =rBhL:

ð8Þ

Displacement u is also related to the reduced height of the hollow section beam, HR ¼H u, participating in the second deformation phase.

171

3.3. Strain rate effect on the energy absorption The blast loading, which causes significant plastic deformations of the hollow beam, is usually associated with high loading rates so that the strain rate effects in mild steel structures have to be taken into account [20]. For example, in the experiments reported in [17], the initial transverse velocities of the analyzed 600 mm long beams vary between 98.3 and 215.8 m s  1 and with the 300 mm long beams analyzed in this study, even higher velocities are associated. Since the model for the local deformation is based on a bending mechanism, the strain rate effects will occur due to the rate of change of curvatures k1 ¼1/R1 and k2 ¼1/R2. The major contribution of the strain rate effects is caused by the variation of curvature k1 ¼1/R1 as it grows rapidly when u increases according to the defined mechanism of deformation. In the present analysis, the strain rate effects on the local deformations are considered in an approximate manner. The energy of the rapidly developed curvature, k1, at the central region of the section, ER1, is partitioned when the equivalent force is obtained as a sum of two forces FðuÞ ¼ F 1 ðuÞ þ F 2 ðuÞ ¼

dER1 dðEnL ER1 Þ þ du du

ð9Þ

where EL is defined by Eq. (3) and ER1 is the bending energy in the central plastic region. The fully plastic bending moment per unit length of the section wall, which determines the energy ER1 , can be expressed as  1=p ! s0 h2 2p hk_ 1 ð10Þ 1þ Md0 ¼ 2p þ 1 2D 4 where k_ 1 ¼ ðdk1 =duÞðdu=dtÞ. The equation of motion (7) for the phase of local deformation now becomes !  1=p ! s0 h2 2p dk1 hu_ mkm ðuÞu€ þ 1þ F 1 ðu=BÞ þF 2 ðu=BÞ ¼ 0 2p þ 1 du 2D 4B ð11Þ For the particular mechanism of local deformation the equivalent forces F1(u) and F2(u) can be best approximated by the following polynomial functions expressed in non-dimensional form F i ðuÞ ¼ ai þ bi ðu=BÞ þci ðu=BÞ2 þ di ðu=BÞ3 þ ei ðu=BÞ4 þ g i ðu=BÞ5 ,

i ¼ 1,2 ð12Þ

2

where F i ¼ 4BF=ðs0 h Þ. A comparison between the equivalent force obtained by Eq. (4) and approximation by Eq. (12) for a strain-rate insensitive material is depicted in Fig. A2(a) in the Appendix, for section 40  40  1.6 mm. The values of coefficients in Eq. (12) are given in Table 1. The plastic deformation is the only mechanism for the energy dissipation so that the plastic deformation energy during this phase is equal to the reduction of the kinetic energy until the entire collapse of the cross-section occurs (u¼ 0.43H at the full collapse of the analyzed square cross-section, i.e., HR ¼0.57H). If no strain-rate effect is taken into account, the reduction of the kinetic energy at u ¼0.43H is equal to the quasi static energy defined by Eq. (1). Fig. A2(b) compares the absorbed energy obtained according to Eq. (1) with the reduction of the kinetic energy obtained by the SDOF model and the two curves are very close, as expected. 4. Global bending It is assumed that the second phase of deformation commences after the full collapse of the hollow section [17–19] and

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while the energy remaining for the global bending of the beam is EG ¼ET  EL. The phase of a global bending of the hollow beam is considered as deformation of a beam of a fixed cross-section, with a reduced height HR and hence reduced plastic moment, M Rp . The wall thickness remains the same. The initial velocity for this phase, V0,G, is obtained from the energy EG as

regarding the beam as the one having a plastic bending moment that varies with the deflection [21,22]. Therefore, the effect of the axial (membrane) force N is taken into account by a modified bending moment Mp-fnMp, where fn ( Z 1) is a membrane factor, which depends on the interaction relationships between M and N [23]. A simplified shape of a fully collapsed section shown in Fig. A3 is assembled in order to estimate the influence of the axial forces on the bending of a hollow beam. It is assumed that the model section has the same cross-sectional area and wall thickness as the original cross-section. The width and the reduced height of the model section as shown in Fig. A3 are obtained as

V 0,G ¼ ð2ðET EL Þ=ðmLÞÞ1=2

B1 ¼ =B2h,

no further distortion of the beam section occurs. An impulsive loading is applied as schematically shown in Fig. 1(a). The total energy, which needs to be absorbed by the beam, is equal to the initial kinetic energy from the blast loading EK ¼ mLV 20 =2 ¼ ET ,

ð13Þ

ð14Þ

where m is mass per unit length of the beam. Due to the high intensity load, plastic hinges develop first at the supports. Following the classical approach for the analysis of an impulsively loaded plastic beam it is assumed that mechanism of deformation is constructed by plastic hinges and rigid segments (Fig. 1(b)). The response history of the global bending consists of two subphases: the first sub-phase, which is characterized by two travelling plastic hinges, and the succeeding sub-phase, with stationary plastic hinges at the supports and the middle of the beam. During the first sub-phase, the travelling hinges propagate from the supports towards the middle of the beam and the hinge velocity is obtained as [3]

x_ ¼ 3ðMSp þ MTp Þ=ðmV 0,G xÞ, 0 r x r l

ð15Þ

M Sp

M Tp

with initial condition x(0) ¼0. and are the fully plastic bending moments at the support and at the travelling hinge, respectively; l ¼L/2. The horizontal middle portion of the beam moves at velocity V0,G and the transverse displacement at the hinge position is W(t)¼V0,Gt. Eq. (15) remains valid until the travelling hinges reach the mid-point of the beam, which is the end of the first sub-phase. In the second sub-phase, the subsequent motion continues with stationary hinges according to the equation € ¼ 3ðM S þM T Þ=ðml2 Þ: W p p

ð16Þ

The initial conditions for this second sub-phase of motion are _ ð0Þ ¼ V 0,G W

ð17aÞ

and Wð0Þ ¼ W I

ð17bÞ

where WI is the displacement at the mid-point of the beam attained at the end of the first sub-phase with travelling hinges. 4.1. Effect of the axial force A method to take into account the influence of the axial force was proposed by Yu and Stronge [21] and this method is adopted in the present analysis. From the viewpoint of energy dissipation, taking account of the contribution of the axial (membrane) force induced by large deflection is equivalent to

BM ¼ H=2:

ð18Þ

The plastic hinge with this idealized cross-section is considered a generalized one, namely, the cross-section of the beam where plastic flow occurs experiences an interaction between the bending moment m ¼M/M0,M and axial force n¼ N/N0, with N0 ¼ 4s0 hðB1 þ BM Þ,

ð19aÞ

M0,M ¼ 2s0 hBM ðB1 þ BM =2Þ,

ð19bÞ

being the fully plastic axial force and the fully plastic bending moment, respectively, of the model cross-section shown in Fig. A3. The parametric representation of the interaction curve when the neutral axis is located between the central line and the lower flange is given by [24] m ¼ 1n2 ð1 þ B1 =BM Þ2 =ð2ðB1 =BM þ 0:5ÞÞ

ð20Þ

while this relationship for the original square hollow section is m¼1 (4/3)n2. The interaction between m and n when the neutral axis is inside the horizontal flange is m ¼ ð1nÞð1 þ B1 =BM Þ=ð1 þ B1 =BM þ 0:5Þ

ð21Þ

while m¼(4/3)(1  n) for the original square hollow section. Noting that the plastic hinges are located at x ¼0 and x ¼ x the rate of elongation of the stretched part is _ =x e_ ¼ W W

ð22Þ

_ =x. The rate and the angular velocity at the hinges is y_ ¼ W of energy dissipated in one half of the beam is  _ M 0,M  W N0 E_ mn =2 ¼ 2M y_ þN e_ ¼ 2m þn W : ð23Þ x M 0,M From the associated flow rule for yield limit (20), the following relationship holds dm N 0 e_ ¼ 2an ¼  , dn M 0,M 2y_

a¼

ð1 þ B1 =BM Þ2 2ðB1 =BM þ0:5Þ

ð24Þ

so that the rate of elongation can be expressed as e_ ¼ 4a

_ M 0,M W n: x N0

ð25Þ

Table 1 Coefficients for the generalized force approximation in the SDOF model (Eq. (12)).

i¼1 i¼2

ai

bi

ci

di

ei

gi

149.6972 8.0551

 1877.2360  6.1226

20.5468 142.0940

 32.2374  2404.3349

1.2737 26.3897

4.0422  127.5680

D. Karagiozova et al. / Thin-Walled Structures 62 (2013) 169–178

which leads to a relationship between the rates e_ and y_ ,

Eqs. (21) and (22) give n¼

N0 W W ¼ q, BM M0,M 4a

173

q ¼ 1=ð1 þ B1 =BM Þ:

ð26Þ

y_ ¼

ðB1 =BM þ0:5ÞN 0 _ e: 2ð1þ B1 =BM ÞM 0,M

ð32Þ

Now, the energy dissipation rate due to bending with the contribution of the axial force can be obtained from Eq. (23) together with Eqs. (20) and (26),  2 ! _ W W E_ mn ¼ , knm,1 ¼ 0:5ðB1 =BM þ0:5Þ 4M 0,M 1 þknm,1 BM x

By substituting the expression for m according to Eq. (24) and using Eq. (30), the energy dissipation rate is obtained merely as a function of the axial force

ð27Þ

Consequently, the following relations for the contribution of the axial force hold ( 1 þ knm,1 ðW=BM Þ2 W=BM o 1 : ð34Þ fn ¼ W=BM Z 1 knm,2 ðW=BM Þ

The energy dissipation rate due to bending only is _ =x E_ m ¼ 4M0:M W

ð28Þ

so that a factor fn  Enm/Em can be referred to as a membrane factor which characterizes the bending moment in the generalized plastic hinge M0,M-fnM0,M, f n ¼ 1 þ knm ðW=BM Þ2

for

ðW=BM Þ o1:

ð29Þ

The interaction between m and n when the neutral axis is inside the horizontal flange is defined by Eq. (21) and in this case, the rate of energy dissipated in one half of the beam can be expressed as   2mM 0,M _ y þne_ : ð30Þ E_ mn =2 ¼ 2M y_ þ Ne_ ¼ N 0 N0 From the flow rule associated with the yield limit (21) the following relationship holds N 0 e_

dm ð1 þB1 =BM Þ ¼ ¼ dn ðB1 =BM þ0:5Þ M0,M 2y_

ð31Þ

_ =x: E_ mm ¼ 2N0 W W Thus, the membrane factor f n ¼ Enm =Em ¼

ð33Þ ðB1 =BM þ 1Þ ðB1 =BM þ 0:5Þ

¼ knm,2 .

For the geometric parameters in this study it turns out that knm,1 ¼ 0.352 and knm,2 ¼1.352. It may be noted that the model section shown in Fig. A3 possesses the same axial force and bending moment as the reduced plastic moment which characterizes the fully collapsed beam section while the model height, BM, in Fig. A3 is slightly smaller than the actual height of the collapsed section, BR (by about 8%). It is observed from the experiments that the cross-sections at the clamped supports are not significantly distorted so that it is further assumed that the bending moment at the supports is equal to the initial plastic moment of the hollow rectangular section, Mp, Mp ¼ s0 BH2 ð1ð12h=BÞð12h=HÞ2 Þ=4

ð35Þ

while the plastic moment associated with the travelling hinges and the stationary hinge at the middle of the beam, which occurs

Fig. 2. Comparison between the characteristics of the deformed square cross-sections obtained from the model and the experiments [17]; (a) partially collapsed section, HR ¼ 0.74H; (a) fully collapsed section, HR ¼ 0.57H; (c) reduction of the section height obtained experimentally: ’—L¼ 300 mm, &—L ¼600 mm.

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D. Karagiozova et al. / Thin-Walled Structures 62 (2013) 169–178

at the second sub-phase of deformation, is equal to the reduced moment M Rp . Thus, the combined expressions, which take into R account the axial force, are taken as Mpfn at the supports and M Rp f n R at the travelling hinges. f n is calculated with respect to the reduced height of the beam section as the new height is HR at the final stage of a local collapse.

combined effect of the axial force and strain rate is taken as 1=p _ =ðDx2 Þ1=P Þ _ MSp ¼ M p f n ð1 þ ðe=DÞ Þ ¼ M p f n ð1 þ ½W W

ð37aÞ

at the supports and R _ =ðDx2 Þ1=P Þ MTp ¼ M Rp f p ð1 þ½W W

ð37bÞ

at the travelling hinges and eventually at the midpoint of the beam. Finally, Eq. (17), which describes the global bending of the beam, becomes

4.2. Strain rate effects

_ =ðDx2 Þ1=p Þ=ðmV 0,G xÞ, 0 r x rl: x_ ¼ 3ðMP f P þ MRp f Rp Þ ð1 þ ½W W

The strain rate effect during the phase of global bending is calculated with respect to stretching. It is assumed that the engineering strain in the segments, which experience stretching, is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 _ =x2 e ¼ ð x þ W 2 xÞ=x  W 2 =2x , i:e: , e_ ¼ W W ð36Þ

The calculated reduced plastic moment, which corresponds to a fully collapsed square section, is obtained as M Rp ¼ 0:513Mp .

where x is the segment passed by the travelling plastic hinge or x ¼L/2 for the stationary hinge at the mid-span. Thus, the

5. Results and discussion

1.2 1

35x35

W/H

0.8

40x40

0.6

50x50

0.4 0.2 0 15

20

25

30

35

40

45

50

I (Ns) 1.4

35x35

1.2

40x40

W/H

1

50x50

0.8 0.6 0.4 0.2 0 25

35

45

55

65

75

85

I (Ns) 1.4 1.2

W/H

1

BC2

0.8

BC1

0.6 0.4 0.2 0 35

40

45

50

55

60

65

70

I (Ns) Fig. 3. Comparison between the model predictions and the experiments for the mid-point deflections of the beams: J—35  35 mm, D—40  40 mm and B—50  50 mm; (a) L ¼300 mm; (b) L¼ 600 mm and (c) influence of the plastic hinge characteristics.

ð38Þ

The average value of the yield stress for the tested hollow sections reported in [17,18] was s0 ¼440 MPa and this value is used further with the model described in Sections 3 and 4. The strain rate effect on the deformation is taken into account with coefficients D ¼40.4 s  1 and p ¼5, which are commonly used for mild steel [25]. Results on blast loading of sections with 1.6 mm thick walls presented in [1] are used to validate the proposed model. A comparison between the experimentally obtained shapes of collapsed sections and those predicted by the present model is shown in Fig. 2((a) and (b)) for a partially and fully collapsed 35  35  1.6 mm sections where a reasonable agreement is observed. The analytically predicted value of the reduced height, HR/H ¼ 0.57 for the square section is compared with several experimentally obtained values for beams of 300 and 600 mm span in Fig. 2(c). The open symbols in this figure correspond to the experimentally observed reduced heights of 600 mm long beams while the closed symbols represent the experimental results for 300 mm long beams. The deflections at the midpoint of the beams as functions of the applied total impulse are presented in Fig. 3(a) for L ¼300 mm and in Fig. 3(b) for L ¼600 mm beams. The symbols J, D, and B are used to mark the experimental results for 35  35  1.6, 40  40  1.6 and 50  50  1.6 mm sections, respectively. Reasonable agreement is observed in Fig. 3(a) for sections 40  40 mm and 50  50 mm while the deflections for 35  35 mm are over-predicted. However, considerable tearing at the supports occurred in the experiments for the latter beams. This indicates that the response type changed to a deformation/failure mode which is not considered by the present model. The comparison between the model predictions and experimental data shows a more consistent agreement for the 600 mm long beams. It is observed that the model tends to predict lower values of the permanent displacements for the smaller impulses while slightly overestimates these values for the larger impulses. This comparison implies the importance of the local deformations. The current model is developed by assuming two strictly sequential phases of deformation, while interaction between the local collapse and global bending occurs in the actual response. Obviously, the beam section does not fully collapse before the global bending commences under blast with relatively small impulses so that the experimentally observed deflections for this range of loading are larger. In contrast, the response to larger impulses is dominated by the global bending when the model behavior is governed by the plastic hinge mechanism. While an abrupt change of the stiffness is modeled by the hinge, a finite hinge length exists in the actual beam and this property is

D. Karagiozova et al. / Thin-Walled Structures 62 (2013) 169–178

10

1.2

40x40

1

8

50x50x1.26mm EG /L (KJ/m)

0.8

EL/ET

175

0.6 0.4

50x50

35x35

40x40x1.6mm

6

32x35x1.85mm

4 2

0.2 0 20

0 25

35

45

55

65

75

85

40

50

60

70

80

I (Ns)

-2

I (Ns)

60

Fig. 4. Proportion of the initial kinetic energy absorbed by the local collapse of the hollow section, L¼ 600 mm.

50x50x1.26mm

50

W (mm)

important for the idealization of the relatively large beam height. The importance of the model assumptions for the hinge mechanism is demonstrated in Fig. 3(c). Boundary conditions BC1 correspond to the mechanism assumed in this study where the hinges at the supports are represented by the initial plastic moment while the travelling hinges are characterized by the reduced plastic moments. The dashed curve in this figure shows the permanent deflections of a beam when a reduced plastic moment is assumed in all the hinges (BC2), which leads to a ‘softening’ of the model. As a whole, the proposed model of dynamic deformation for the hollow beam describes the overall response appropriately. It should be also noted that, due to the large deflections of the beam, significantly large impulses cause more severe deformation of the cross-section (especially in the lower flange) than that assumed in the model. This behavior possibly contributes to an increase of the energy absorption by the section alone thus leading to smaller permanent displacements observed in the experiments in comparison with the model predictions. The above results show that two characteristic features of the blast response of the hollow section beam distinguish the response of this structural component from the response of its solid counterpart. First, the impulsive loading is applied to the upper flange of the beam, which has a considerably lower mass than the entire beam. As it follows from the momentum conservation, the attained initial velocity of the upper flange is considerably higher than when the total mass of the beam section is taken into account. It leads to the fact that the corresponding kinetic energy is higher too, because the kinetic energy is proportional to not only mass and but also velocity squared. Therefore, a larger input energy must be absorbed by the plastic deformations of the hollow beam during the response. On the other hand, the preceding local deformation phase can reduce significantly the amount of kinetic energy which needs to be absorbed by the global bending of the beam. The outlined characteristics of deformation verify the results for the final midpoint deflections presented in Fig. 3((a) and (b)). One can see that the curves, which show the trend of the displacements, do not pass through the origin of the coordinate system as relatively small impulsive loads can be entirely dissipated by the local deformation of the beam section without causing global deformations. Namely, a threshold value of impulse exists for the presence of global bending. Fig. 4 shows the proportion of the local energy, EL/ET, which can be absorbed by the full collapse of the section, with respect to the total energy in the beam. It is seen that as the impulse increases, the total energy imparted onto the beam increases and more global deformation has to take place. Hence the fraction of local energy

30

40x40x1.6mm

40

32x35x1.85mm 30 20 10 0 20

30

40

50

60

70

80

I (Ns) Fig. 5. Influence of the section wall thickness for equal mass hollow beams, L¼ 600 mm; (a) energy remaining in the beam after the local collapse of the hollow sections and (b) final deflection at the midpoint of the beams.

dissipated in phase one decreases. An upper limit of the impulse can be defined, below which the initial energy can be absorbed merely by the local collapse of the section. The influence of the mass distribution in the hollow section is discussed next, based on the comparison of the responses of three beams of square section with equal total mass but having different combinations of side and wall thickness—beam A: 35  35  1.85 mm; beam B: 40  40  1.6 mm and beam C: 50  50  1.26 mm. The beams are subjected to blasts with equal total impulses. Beam A is the thickest (also smallest) and has the largest proportion of the input energy of the beam absorbed by the local deformations, because this part of the energy increases proportionally to the section thickness as h2. Therefore, the energy available for the global bending of this beam is the smallest (Fig. 5(a)). On the other hand, the initial and reduced plastic moments of beam A are the smallest among the analyzed beams (Mp,A ¼1343.17 Nm, Mp,B ¼1558.03 Nm and Mp,C ¼1975.98 Nm) so that the difference in the final deflection among the three beams reduces for the higher impulses as shown Fig. 5(b). One can see that the height of the beam section plays an essential role for relatively small impulses when a significant part of the input energy can be absorbed by the local collapse of the section. For large impulses, however, the final deflections differ less, which indicates that deformation becomes dominated by the global bending. Instead of square sections, hollow rectangular sections with different sectional aspect ratios (H/B), are another type of equal mass beams. The responses of two beams, one with a square section 35  35  1.6 mm and the other having B ¼40 mm and H¼30 mm and h¼ 1.6 mm, are compared in Fig. 6. It is assumed that after the full collapse of the rectangular section shown in Fig. 6(a) no further distortion of this section develops. Two loading conditions are analyzed—when the beams are subjected

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Fig. 6. Comparison between the deformation of equivalent mass square section, 35  35  1.6 and 30  40  1.6 mm rectangular section; (a) fully collapsed rectangular section; (b) EG/L, blast loading with equal total impulses; (c) EG/L, blast loads with equal impulses per unit area and (d) ratios of the mid-point deflections.

1.8

35x35 40x40

1.5

W/H

1.2

50x50

35x35 40x40

0.9

50x50

0.6 0.3 0 25

35

45

55

65

75

85

I (Ns) Fig. 7. Strain rate effects on the beam response, L¼ 600 mm (solid curves: model predictions without strain rate effects; dashed curves: model predictions with strain rate effects).

to equal total impulses and when they are subjected to equal impulse per unit area of the upper flange. The energy absorption during the local deformation phase is calculated in the same way as for the square section when a reduced height HR ¼0.6H of the rectangular section is obtained. The reduced plastic moment of the rectangular section is calculated as M Rp ¼ 0:6615 M p where Mp ¼ 1052.56 Nm. The energy remaining in the beams for the phase of global bending is compared in Fig. 6(b) for impulsive loading with equal total impulse. In this case, the attained kinetic energy in the square section beam is larger than that in the rectangular section beam. Due to the larger energy absorption capacity of the square section, however, a more significant reduction of EG can be achieved in this section. Therefore, the energy available for the global bending phase still can be smaller than the one in the rectangular section for relatively small impulses. Due to this mechanism of energy absorption, the final deflections of the square section beam are smaller than those in the rectangular section for small impulses. For larger impulses, the larger total energy in the square section cannot be reduced sufficiently in order to maintain it lower than in the rectangular beam. Nevertheless, the final deflections in both

kinds of hollow beam remain comparable for the larger impulses due to the larger bending resistance of the square section (M Rp ðSQ Þ ¼ 799:25 Nm, while M Rp ðRECÞ ¼ 704:5 Nm) despite the larger energy EG in the beams with square section. An equal impulse per unit area causes a significantly larger kinetic energy in the rectangular section (Fig. 6(c)), because of the larger face area exposed to the blast. The combination of the lower energy absorption capacity of the rectangular section and its lower bending resistance leads to larger final deflections than the square section. The final deflections of the rectangular section beam are significantly larger than those of the square section beams particularly for small impulses. The ratios WR/ WSQ are shown in Fig. 6(d) for the two loading cases. The deflection of the square section is always smaller than that of a rectangular section with larger side facing the blast. It is possible that if the smaller side of a rectangular section is subjected to blast loading, the result may be different. The present analysis demonstrates that the strain-rate effects do not change qualitatively the response of the hollow section but the increase in the material strength at higher strain rates leads to smaller final deflections. The model predictions of the final deflections with and without strain rate effects are compared in Fig. 7 (the solid curves represent the results without the strain rate sensitivity effects). It should be noted that the energy dissipated by the section collapse increases significantly, thus leading to an increase of 65% to 76% in the impulse that can be absorbed entirely by the local collapse of the beam section. The above analysis of the behavior of hollow section beams suggests that, unlike the predictions for the response of a solid section beams, the commonly used ratio between the final deflections and the initial height of the beam section does not provide sufficient information on the beam behavior. The response of a hollow section beam cannot be characterized by a simple relationship between the impulse and deflection due to the presence of two-phase deformation process. For this reason, the initial conditions for an impulsive loading caused by blast loadings cannot be obtained when the total mass of the beam section is used. It appears that this type of response of the hollow beam resembles somehow that of a sandwich beam, where similar to the energy absorption due to the local collapse of a thin-walled section, the core is able to absorb a significant proportion of the blast energy.

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bending rigidity when considering the reduced plastic moment of the beam and taking into account the effect of axial force. The methodology established in developing the present analytical model may be applied to studies of dynamic response of other similar beams subjected to blast loading.

It should be emphasized that the present model was developed when no interaction between the two phases of deformation was assumed. A non-negligible influence of the coupling between the two phases could occur for stiffer sections with smaller slenderness ratios B/h when a full collapse of the cross-section cannot be reached before the global bending has commenced. In this case, the response of the hollow section beam will be characterized by the global deflection even for small impulses as observed in [16] and a more complex model that takes into account the interaction between the two deformation phases should be developed.

Acknowledgements The work was conducted when the first and second authors visited NTU; the financial support from Defence Research and Technology Office and NTU is gratefully acknowledged. The third author (G. Lu) acknowledges the support from NSFC (No. 11028206) and the State Key Laboratory of Explosion Science and Technology (Beijing Institute of Technology) (project KFJJ11–1M).

6. Conclusions A two-phase deformation model of a hollow section beam under a blast loading is developed in order to reveal the characteristic features of deformation and energy absorption of hollow section beams under a near-field blast. The proposed model can describe the temporal variation of the local and global deformations when the strain rate effects are taken into account during both the phases of deformation. Reasonable agreement is obtained in comparison with the experimental data published in the literature [17,18]. It was established that two characteristic features of the response of a hollow section distinguish the response of this structural component from the response of its solid counterpart. First, the initial kinetic energy generated by the blast is considerably higher as the impulsive load is imparted on the upper flange of the beam, which has a significantly lower mass than the entire member. Therefore, a larger amount of energy must be dissipated by the plastic deformations during the response. On the other hand, the preceding local deformation phase can significantly reduce the amount of remaining energy which needs to be absorbed by the global bending of the beam. It has been shown that the mass distribution in the hollow section is an important factor in determining the energy partitioning between the local deformation phase and global bending of the beam. The energy dissipated during the local collapse plays an essential role as it absorbs a significant proportion of the initial kinetic energy generated by the blast. There is an upper limit of the blasts that can be absorbed entirely by the section collapse leaving no energy for global bending of the beam. The response of a hollow beam to large blasts is dominated by the beam

Appendix. Geometric relationships for the local collapse of the beam section The local collapse of the beam cross-section is determined by the vertical displacement u which is measured at the highest point. From the geometry of the section the following relationships hold

Dk1 ¼ 1=R1

ðA1aÞ

Dk2 ¼ 1=R2

ðA1bÞ

b ¼ ga

ðA1cÞ

where R1 ¼ ðB=2lÞ=g

ðA2aÞ

__ R2 ¼ M F =a

ðA2bÞ __ 1 For PM oB=2 the arc length M F ¼ uð1a sin aÞ and the angles a and g are related as __ R1 sin g þlcos g ¼ B=2M F a1 sin atanða=2Þ ðA3Þ

Thus, for a given u, the deformation energy EnL ðuÞ ¼ minja, bEnL ðu, a, bÞ . The minimum energy is obtained for a ¼ g when the displacements u cause a bending of arc with associated PM o B=2. __ For PM ¼ B=2 the arc length M F ¼ u=ð1a1 Þ, angle g remains constant and the energy increases due to the further bending of

F

u O1 γ

β D

R1 C

α O2

R2

HR Q

P

H

M

B/2

Fig. A1. (a) Experimentally observed deformed shape of a square section [17] and (b) Model for the local deformation of the cross-section.

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References

3

E*L (KJ/m)

2.5 2

Eq (1) 1.5

SDOFmodel

1 0.5

u = uc 0 0

0.005

0.01

0.015

0.02

u (m) 50

F(u/B)

40 30 20

Best fit approximation

Eq (9)

10 0 0

0.1

0.2

0.3

0.4

0.5

u/B Fig. A2. Equivalent force (a) and deformation energy for the local section collapse (b) due to the deformation mechanism and according to the SDOF model; section 40  40  1.6 mm, s0 ¼ 440 MPa.

B

H

BM

B1 Fig. A3. Schematic model for obtaining the membrane factors (Eq. (34)) of the generalized plastic hinge.

__ arc M F when a satisfies the equation uð1cos aÞ1 ¼ ð1a1 ÞB=2

ðA4Þ

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