Steel square hollow sections subjected to transverse blast loads

Thin-Walled Structures 53 (2012) 109–122

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Steel square hollow sections subjected to transverse blast loads H.H. Jama a, G.N. Nurick b, M.R. Bambach c,n, R.H. Grzebieta c, X.L. Zhao a a b c

Department of Civil Engineering, Monash University, VIC 3800, Australia Blast Impact and Survivability Research Unit (BISRU), Department of Mechanical Engineering, University of Cape Town, Private Bag Rondebosch 7701, South Africa TARS, Faculty of Science, University of New South Wales, Old Main Building (K15), Sydney, NSW 2052, Australia

a r t i c l e i n f o

abstract

Article history: Received 30 October 2011 Received in revised form 10 January 2012 Accepted 10 January 2012 Available online 7 February 2012

Thin-walled steel hollow sections are used extensively in the construction, offshore, mining and security industries. Such members subjected to blast loads are of interest due to increased security demands and the occurrence of accidental or intentional explosive events. This paper reports an experimental and analytical investigation of steel square hollow sections subjected to transverse blast load, applied with explosive uniformly distributed along the length of the member. Three different section sizes were tested over three different lengths, and the members were fully clamped at their ends. The explosive loads were sufficient in magnitude to cause plastic deformation of the cross-section (local deformation), plastic flexural deformation of the overall member (global deformation), and tensile tearing at the supports. The energy dissipated in the local deformation is determined using rigid-plastic analysis and yield line mechanisms of the deformed cross-sections. The total input energy minus the energy dissipated in local deformation is assumed to be expended in flexural deformation. Analytical solutions using the energy consumed in flexural deformation are shown to produce bounded solutions of the transverse plastic deformation of the members. Finally, a semi-empirical solution is suggested that can be used to aid in design. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Steel tubular sections Blast load Explosive load Rigid-plastic theory Impulse

1. Introduction Structural members subjected to blast loads are of interest due to increased security demands and the occurrence of accidental or intentional explosions. Experimental investigations of solid metal beams subjected to impulsive loads began in the 1960s [1–4], where plastic flexural deformations were investigated. Menkes and Opat [5] extended these works to describe the three fundamental deformation modes of; large inelastic deformation (Mode I), tensile tearing at the supports (Mode II) and transverse shear failures at the supports (Mode III). Experimental works then began to investigate different structural assemblies and sections, including frames [6,7], T-sections [8], hollow sections [9,10], plates [11,12] and composites [13,14]. Many of these experimental works used ballistic pendulums to determine the magnitude of the explosive load [2,4,8–14], and the long history of this technique has proved it accurate and reliable. There has similarly been a long history in the development of analytical solutions for metal structural members subjected to transverse blast loads. Analytical solutions broadly fall into two categories; Single Degree of Freedom (SDOF) and the plastic analysis

n

Corresponding author. Tel.: þ61 2 9385 6142; fax: þ 61 2 9385 6040. E-mail address: [email protected] (M.R. Bambach).

0263-8231/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2012.01.007

approach. The SDOF approach provides the fundamental response of a structure subjected to a dynamic load and uses the equivalent mass, damping and resistance functions to estimate the response [1]. Baker [15] used Single Degree of Freedom (SDOF) analysis to predict the tip deformations of a cantilever subjected to a blast load. Baker et al. [16] similarly showed the use of SDOF approximations to yield satisfactory results when compared to the experiments of Florence and Firth [4]. SDOF analysis is particularly favoured when coupled with a pressure–impulse diagram to yield an iso-damage curve. Such an iso-damage curve, usually called a Pressure–Impulse (P–I) curve, can be used to determine whether a structure subjected to certain loading is safe or not. However, while the SDOF approaches have been found to be useful in the analysis of structural elements, they may become difficult to use in structural systems [17]. This is especially true when multiple modes of deformation occur, which potentially makes the use of the SDOF approach non-conservative. This is similar to the local and global deformations that occur in the steel hollow sections in the present paper, which makes the SDOF approach not applicable. Plastic analysis is based on the assumption that the structure is subjected to loads that are much higher than could be absorbed in a wholly elastic manner. Humphreys [2] presented analytical solutions of steel beams subjected to impulsive loads which disregarded the effects of both strain hardening and strain rate hardening. Florence and Firth [4] used rigid-plastic analysis

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including strain hardening and finite displacements to predict the final deflections of pinned and clamped beams subjected to impulsive loads. They suggested that the inclusion of strain rate hardening would improve their prediction of the final midspan deformation of the beams. Symonds and Chon [18,19] used bound methods to approximate the mode shapes and final deformations of structures subjected to impulsive loads. Jones [20] demonstrated the use of rigid-plastic analysis of beams and plates subjected to impulsive loads including the influence of finite deflections. Jones and Oliviera [21] suggested that rotary inertia plays ‘‘a small but not negligible’’ role on the response of beams subjected to impact loads. Symonds and Jones [22] showed the importance of including strain-rate effects to rigid-plastic solutions when applied to mild steel beams. Jones [23] presented his seminal work on the failure modes of beams subjected to impulsive loads, where the final deflection and transition velocities for Modes I and II deformations, as defined by Menkes and Opat [5], were solved analytically. He also showed the onset of

Mode III failure to be dependent on the initial velocity of the beam, which is analogous to the impulse imparted to the beams. Other notable works investigated the effects of shear on transversely blast loaded members. Hodge [24] showed that for beams where the length is large in relation to the beam depth the effect of shear is negligible. As the beam length is reduced with respect to beam depth, the effect of shear on the response of the beam becomes important. Similarly, Karunes and Onat [25] suggested that shear will dominate bending when the value QoL/Mo is less than 20 in a ductile beam subjected to an impact at the midpoint, were Qo and Mo are the plastic shear and moment capacities, respectively. This is similar to the conclusion drawn by Jones [26] that shear can be neglected on beams subjected to impulsive loads provided QoL/Mo is less than 15. Drucker [27] showed the influence of shear on the response of cantilever beams through the use of interaction curves. It was shown that a small amount of shear produces a second-order reduction in the limit moment of the beams. Bleich and Shaw [28] demonstrated

Table 1 Experimental specimens and deformation results (specimens 20 to 31 were excluded from the present paper). Test no.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53

Section bHt (mm)

35  35  1.6 35  35  1.6 35  35  1.6 35  35  1.6 35  35  1.6 35  35  1.6 40  40  1.6 40  40  1.6 40  40  1.6 40  40  1.6 40  40  1.6 40  40  1.6 50  50  1.6 50  50  1.6 50  50  1.6 50  50  1.6 50  50  1.6 50  50  1.6 50  50  1.6 35  35  1.6 35  35  1.6 35  35  1.6 35  35  1.6 35  35  1.6 40  40  1.6 40  40  1.6 40  40  1.6 40  40  1.6 50  50  1.6 50  50  1.6 50  50  1.6 50  50  1.6 50  50  1.6 35  35  1.6 35  35  1.6 40  40  1.6 40  40  1.6 40  40  1.6 50  50  1.6 50  50  1.6 50  50  1.6

Span Mass/ (mm) metre (kg/m)

300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 600 600 600 600 600 600 600 600 600 600 600 600 600 600 1000 1000 1000 1000 1000 1000 1000 1000

1.63 1.63 1.63 1.63 1.63 1.63 1.88 1.88 1.88 1.88 1.88 1.88 2.38 2.38 2.38 2.38 2.38 2.38 2.38 1.63 1.63 1.63 1.63 1.63 1.88 1.88 1.88 1.88 2.38 2.38 2.38 2.38 2.38 1.63 1.63 1.88 1.88 1.88 2.38 2.38 2.38

Quasistatic Ultimate stress (MPa)

(%)

(le)

430 430 430 430 430 430 430 430 430 430 430 430 410 410 410 410 410 410 410 391 391 391 391 392 384 384 384 384 428 428 428 428 428 444 491 403 403 403 421 421 421

495 495 495 495 495 495 495 495 495 495 495 495 475 475 475 475 475 475 475 471 471 471 471 471 440 440 440 440 456 456 456 456 456 491 468 456 456 456 482 482 482

27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 23.7 23.7 23.7 23.7 23.7 24.6 24.6 24.6 24.6 23.2 23.2 23.2 23.2 23.2 19.2 21.2 23.5 23.5 23.5 22.9 22.9 22.9

26.1 26.1 26.1 26.1 26.1 26.1 30.2 30.2 30.2 30.2 30.2 30.2 37.5 37.5 37.5 37.5 37.5 37.5 37.5 24.9 24.9 24.9 24.9 24.9 28.5 28.5 28.5 28.5 38.3 38.3 38.3 38.3 38.3 26.5 27.9 29.2 29.2 29.2 38.0 38.0 38.0

M ¼Molar shape and T ¼ Tear drop shape (Fig. 4). a b

Eq. (1). Fig. 4.

Strain Slenderness ratioa at failure

Quasistatic Yield stress (MPa)

No. strips Mass of of explosive explosiveb (g)

2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 3 3 3 3 3 1 1 1 1 1 1 1 1

16.6 21.6 26.1 29.5 31.7 37.5 16.6 21.6 26.2 29.0 32.0 36.0 11.9 16.9 22.0 26.6 29.0 32.0 36.0 22.0 27.0 32.0 37.0 42.0 22.0 27.0 33.0 37.0 33.0 39.0 42.0 51.0 63.0 51 61 51 61 71 51.0 61.0 72.0

Impulse (Ns)

25.7 35.5 42.0 42.2 50.6 58.7 25.7 26.6 35.6 41.3 48.4 58.2 19.3 27.3 35.8 42.2 47.7 53.5 60.0 39.1 49.16 56.04 58.32 66.77 38.96 49.06 61.43 64.65 47.78 67.12 72.62 76.15 94.07 107.7 125.4 90.1 107.7 125.4 90.1 107.7 125.4

Global Local Local deformation deformation deformed shape (delta/H)

(delta/H)

0.23 0.43 0.62 0.83 0.91 1.06 0.24 0.31 0.50 0.62 0.87 1.05 0.11 0.24 0.40 0.45 0.53 0.63 0.72 0.52 0.78 0.92 0.79 0.97 0.30 0.45 0.71 0.64 0.41 0.56 0.64 0.81 1.01 1.07 1.13 0.67 0.89 1.25 0.47 0.49 0.70

0.80 1.00 1.17 1.20 1.26 1.26 0.85 0.95 1.15 1.20 1.24 1.28 0.56 0.72 0.9 1.04 1.16 1.22 1.32 0.86 1.06 1.03 1.11 1.20 0.75 0.95 1.10 1.10 1.18 1.12 1.30 1.34 1.34 0.86 0.71 0.75 0.80 1.08 0.83 0.91 1.06

M T T T T T M T T T T T M T T T T T T M M M T T M M T T T T T T T M M M M T M M T

Global failure mode

I I I II* II* II I I II* II* II* II* I I I II* II* II* II* I I I I I I I I I I I I I I I I I I I I I I

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that shear stresses dominate bending stresses in the early stages of a beam subjected to a large impulsive load. Neal [29] found the influence of shear and normal forces on the bending moment of a beam of rectangular cross-section and presented an interaction curve for shear, normal and bending forces. Nonaka [30] showed the influence of bending and shear on beams subjected to impulsive loads. All the aforementioned impulsive studies dealt with beams, frames or plates with solid cross-sections, except those in [9] of a single size of steel tube, and [10] of aluminium tubes. The behaviour of steel hollow sections are well known to be highly dependent on the section slenderness (non-dimensionalised width to thickness ratio of the tube wall). The present paper presents the results of an experimental investigation of 43 steel hollow members subjected to transverse blast load. Three different section sizes were tested to produce a range of slenderness ratios, and three different member lengths were tested to determine the length effect, on the local and global deformations. The experiments were performed on a ballistic pendulum located at the Blast Impact and Survivability Research Unit (BISRU), at the University of Cape Town. The failure modes and deformations are compared with established rigid-plastic solutions, which are shown not to be generally applicable since they neglect the local cross-section deformations. A plastic yield line mechanism approach is used to model the local deformations based on the experimental deformed shapes, and the energy dissipated in locally collapsing the cross-sections is determined. The remaining energy is assumed to be consumed in global flexural deformation, and is shown to provide bounded solutions for the experimental global flexural deformations. Finally, a semi-empirical solution is suggested that can be used to aid design.

111

pendulum. The specimen dimensions and slenderness ratios (Eq. (1)) are summarised in Table 1. It is noted that specimen numbers 20 to 31 have been excluded from Table 1, since they had non-fixed boundary conditions and are not applicable to the present paper. The section slenderness ratio is given by: rffiffiffiffiffiffiffiffiffi sy b2t le ¼ ð1Þ t 250 where b is the flat width, t is the thickness and sy is the yield stress of the hollow section. 2.2. Ballistic pendulum The ballistic pendulum shown in Fig. 1 was used to measure the applied impulse for all members, except those that were of length 1000 mm. The 1000 mm length members could not be accommodated on the pendulum, and were tested in a fixed frame with the same end restraints. The applied impulse for these members was determined from the known relationship between the explosive mass and resulting impulse, established with the ballistic pendulum from the 300 mm and 600 mm length tests. The ballistic pendulum is essentially an I-beam suspended from the roof using two pairs of spring steel wires. At one end there is a flat plate onto which a base plate and a pair of clamps were fixed. On the other side, there are a number of counter weights to achieve balance. A pen and a paper were used for recording the maximum displacement of the pendulum once the explosion had been initiated. The velocity of the pendulum can be determined from the oscillations and geometric properties of the pendulum. The impulse is then given by: I ¼ M x_

ð2Þ

2. Experimental details

where M is the mass and x_ is the velocity of the pendulum.

2.1. Member dimensions

2.3. Application of explosive

The members were commercially available cold-formed steel square hollow sections (SHS) produced in Australia and shipped to BISRU for testing. Three different sizes of SHS were tested, with nominal dimensions of 35 mm, 40 mm and 50 mm and nominal thickness of 1.6 mm. Three member lengths of 300 mm, 600 mm and 1000 mm were tested. The member ends were rotationally and axially restrained using steel blocks inserted inside the tube ends, which were then bolted to the rigid supports of the ballistic

The members were approximately uniformly loaded using PE4 plastic explosive in one, two or three equally spaced strips as shown in Fig. 2, depending on the width of the cross-section and the mass of the explosive. The explosive was laid onto a 13-mm-thick polystyrene foam pad to prevent local spalling of the steel, and to provide a stand-off distance and hence attenuate and uniformly distribute the applied impulse. The explosive strips were made by rolling PE4 explosive into cylinders and arranging in such a way that

Ceiling

Spring steel wires

Strenghtening beams

Counter balancing masses

Base plate Support beam 12mm

Base plate

Span

Detonator Specimen

Tube 500 I- beam Tracing paper Clamps

Recording pen

Floor SIDE ELEVATION

1000 mm FRONT ELEVATION

Fig. 1. Schematic of the experimental setup on the ballistic pendulum.

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H.H. Jama et al. / Thin-Walled Structures 53 (2012) 109–122

L I

13 mm

Explosive b/2

b Detonator

Specimen Polystrene atenuator

13 mm

Explosives b/4 b Detonator

Polystrene atenuator

Specimen

13 mm

Explosives b/4 b

Detonator Fig. 2. Schematic of the application of the explosive. (a) 1strip, (b) 2 strips and (c) 3 strips.

500 450 400

300

Stress (MPa)

Stress (MPa)

350

250 200 150 100 50 0 0

0.05

0.1 0.15 Strain [ - ]

0.2

0.25

800 750 700 650 600 550 500 450 400 350 300 250 200 150 100 50 0

82/s 174/s 314/s 403/s 0

0.02

0.04

0.06 0.08 Strain (-)

0.1

0.12

0.14

Fig. 3. Typical stress–strain curves of the tested SHS; (a) quasi-static, (b) high strain rate.

there was approximately a uniform distribution of explosive mass over the specimen. The parallel strips of explosive were connected at the midspan by cross-leaders. A short tail of 1 g of explosive holding the detonator was then attached to the centre of the cross-leaders. The masses of the explosive tail and the cross-leaders were kept at 1 g each for all the tests. A similar technique of creating a uniform impulsive load using plastic explosive has been used in previous experiments [8–14].

2.4. Material properties of the SHS Standard tensile coupons were cut from the SHS, and quasistatic tensile tests were carried out at strain rate of 2.5  10  3 s  1 using an Instron 4204 testing machine. A typical stress–strain curve is shown in Fig. 3a. The results of the yield stress (0.2% proof stress), ultimate stress and strain at fracture and the average values are listed in Table 1.

H.H. Jama et al. / Thin-Walled Structures 53 (2012) 109–122

113

2L

global

local

H

H

local

Fig. 4. Measurement of local cross-section and global flexural deformations. (a) Measurement of global deformation, (b) measurement of local deformation when top flange does not hit bottom flange – "Molar shape" and (c) measurement of local deformation when top flange hits bottom flange – "Tear drop shape".

Additional specimens were machined from a sample of the SHS and high strain rate testing was carried out at the Structural Impact Laboratory (SIMLab) of the Norwegian University of Science and Technology (NNTU), using a Split-Hopkinson Pressure Bar (SHPB) in tension [31]. The flat specimens were glued using araldite to machined holders which were then screwed onto the SHPB. Shock loading was achieved by breaking a friction lock and tensile waves propagated through the specimens. The purpose of the tests was to determine the Cowper–Symonds strain rate parameters most suitable for cold-formed steel, which provides the relationship between dynamic yield strain and static yield strain: n¼

 1=p

sd e_ ¼ 1þ sy D

ð3Þ

where n is the strain rate index, sd is the dynamic yield stress, sy is the static yield stress, e_ is the strain rate and D and p are the Cowper–Symonds strain rate parameters. Strain rates of between 82/s and 403/s were obtained during the high strain rate testing. A sample of the results is presented in Fig. 3b. There is disagreement in the literature regarding the appropriate values of the Cowper–Symonds strain rate parameters D and p that should be used for mild steel. The widely used parameters D ¼40.4 and p ¼5 were originally proposed by Cowper and Symonds [32], based on the experimental work of Manjoine [33] as reported by Bodner and Symonds [1]. In contrast, Yu and Jones [34] proposed values of D ¼1.05  107 and p¼ 8.3, while Abramowicz and Jones [35] used D ¼6844 and p¼ 3.91 which was based on the work of Campbell and Cooper [36]. The differences result in part from the different chemical contents of the materials tested, heat treatments of the specimens, and strain ranges over which the parameters were evaluated. Comparison of Eq. (3) with the present SHPB results showed that the values of D ¼884 and p ¼2.207 as published by Marais et al. [37] are suitable for the cold-formed steel of the SHS. It is noted that these results were obtained using the stress corresponding to 2% strain, as there were considerable difficulties in obtaining the ideal yield stress at 0.2% strain with good accuracy.

3. Test results 3.1. Experimental observations The relationship between the impulse recorded by the ballistic pendulum and the mass of applied explosive was found to be highly linear, with a coefficient of determination of 0.96 [38]. This relationship was used to determine the impulse for the 1000 mm length members that were not tested on the ballistic pendulum. As a result of the transverse blast load, the members underwent global and local permanent deformations. Global deformations refer to the overall flexural deformation and local deformation refers to the local cross-section deformation of the members. The global deformations were measured as shown in Fig. 4a and refer to the maximum permanent displacement of the bottom flange at the mid-length. The local deformations were complex and the measurement of the local deformation was taken to be the distance traversed by the top flange. The measurement of the local deformation depended on whether the top flange (loaded side) had come into contact with the bottom flange and is shown in Fig. 4b and c. It was observed that as the impulse on the members was increased, the deformed cross-section changed from a ‘‘Molar’’ to a ‘‘Tear-drop’’ shape. 3.2. Deformation of 300 mm length members—modes I and II The experimental deformations, shapes and modes are summarised in Table 1, and sample photographs of the final deformation of the 40 mm sections with 300 mm span when subjected to increasing impulse are shown in Fig. 5a. These photographs are also representative of the deformations observed in the 35 mm and 50 mm sections with a 300 mm span. Fig. 5a shows the side view of the members, and adjacent is the cross-section at the mid-length of the same member. The top flange collapsed along the length while the webs folded inwards. At lower impulses, the top flange undergoes large inelastic deformation while the bottom flange remains straight and the cross-section forms a ‘‘Molar’’ shape. At higher impulses, the top and bottom flanges come into contact with each other and form a ‘‘Tear drop’’ shape. For the 40 mm sections, the contact between the top and bottom flanges

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H.H. Jama et al. / Thin-Walled Structures 53 (2012) 109–122

Fig. 5. (a) Photographs of a sample of 40  40  1.6 mm SHS of length 300 mm (from the top: specimens 7, 8, 9, 10, 11, 12—Table 1) (b) Photographs of a sample of 35  35  1.6 mm SHS of length 600 mm (from the top: undeformed specimen, specimens 32, 33, 34, 35, 36—Table 1). The 600 mm length specimens were cut at the midspan prior to the photograph in Fig. 5b being taken. The photographs on the right side show the cross-section at the midspan.

occurs at 35.50 Ns. A similar failure pattern of the cross-section was observed in the 35 mm and 50 mm sections. The onset of Mode II (tearing) was observed at an impulse of approximately 42 Ns for the 35, 40 and 50 mm sections. It was reported by Jones [23] that the threshold of Mode II failure for beams with a solid cross-section depended on the impulse imparted onto the beam and the beam depth. It appears the tearing threshold of the 35 mm, 40 mm and 50 mm sections is governed by the thickness of the top flange. All the sections had a wall thickness of 1.6 mm and therefore the threshold of Mode II failure is similar. More II deformation was initially defined by Menkes and Opat [5] as referring to tensile tearing at the supports. In [5] tensile tearing meant complete tearing of the beams from the supports. However, Mode II failure is complex, and in plates partial tearing along the clamped boundary has been reported. Mode II failure was categorised by Olson et al. [39] as Mode II-1, II-2 or II-4 indicating tearing along the boundary of a square plate on one, two or four sides respectively. Mode II deformation was classified by Nurick et al. [12] as Mode II and Mode IIn in circular plates which refer to complete and partial tearing along the clamped boundary respectively. Based on the reclassifications of Nurick et al. [12], the deformation of the hollow square steel members subjected to blast loads may be viewed as Mode IIn, since they tear partially at the supports. There was one instance in the 35 mm sections when there was complete tearing from the boundary and this is classified as Mode II and is indicated in Table 1.

3.3. Deformation of 600 mm length members—mode I The experimental results are tabulated in Table 1, and Fig. 5b shows a sample of the 40 mm sections with 600 mm span. The 600 mm length specimens were cut at the mid-length prior to the photograph in Fig. 5b being taken, in order to analyse the posttest deformation. Similar to the 300 mm span members the shapes changed from ‘‘Molar’’ to ‘‘Tear drop’’ with increasing impulse (Table 1), however only Mode I deformations were recorded. Explosive masses that would be required to reach Mode II deformations for members of this size could not be applied due to the limitations of the experimental facility. 3.4. Deformation of 1000 mm length members—mode I Results for the 1000 mm sections are summarised in Table 1 and were similar to the 600 mm members, where the limitations for the explosive mass resulted in only Mode I failures. This limitation also resulted in predominantly ‘‘Molar’’ shape responses, with few ‘‘Tear drop’’ shapes.

4. Comparison of the experimental results with existing theories 4.1. Theoretical solutions The solution derived by Jones [23] for predicting the global transverse deformation at the mid-span of a fixed-ended metal

H.H. Jama et al. / Thin-Walled Structures 53 (2012) 109–122

such as steel, and was included using Eq. (6): " # 3=2 1=p H2 cl n ¼ 1þ 48L3 D

beam with a solid cross-section, including finite deformations that occur in axially restrained beams, is given by Eq. (4):

d H

¼ 0:5ð

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ 0:75lÞ1Þ

ð4Þ

mV 2o L2

ð5Þ

Mp H

r

where c is the speed that shockwaves travel through the material, and for steel an approximate value of 5000 m/s is widely accepted. To evaluate the global flexural deformation of a mild steel beam with strain rate effects, the yield stress used to calculate the plastic bending moment in Eq. (5) is replaced with the dynamic value, determined by increasing the static value by n (Eq. (6)). This results in the use of Eq. (8) in place of Eq. (5):

where m is mass/unit length of beam, Vo is the initial velocity, L is one half-span of the beam, and M p is the plastic moment capacity of the beams. Eq. (4) was shown by Jones [23] to provide bounding solutions to the experimental data of Menkes and Opat [5] when inscribing and circumscribing yield criteria are used. To obtain upper bound solutions, the yield stress used to calculate the plastic bending moment in Eq. (5) is replaced with 0.618sy. The inclusion of the strain-rate sensitivity of the material was shown to be important for metals that display such sensitivity,

Global midspan deformation Delta/H [ - ]

4.50

l¼

mV o 2 L2

ð8Þ

nMp H

35mm Exp 40mm Exp 50mm Exp Bounds - Equation 4 with strain rate Bound - Equation 4 without strain rate

4.00 3.50 3.00 2.50 2.00 1.50 1.00 0.50 0.00

10

0

20

40

30

50

70

60

Vo2L2 nM0H

Global midspan deformation Delta/H [-]

4.0 35mm Exp 40mm Exp 50mm Exp Bounds _ Equation 4 with strain rate Bounds - Equation 4 without strain rate

3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 0.0

2.5

5.0

7.5

10.0

12.5

ð6Þ

where H is the depth of the beam, l is as previously defined, L is one half-span of the beam, D and p are Cowper–Symonds strain rate parameters and c is obtained thus: sffiffiffiffiffi fy ð7Þ c¼

where d is the midspan permanent deformation, H is the section depth and l is a dimensionless load and geometry factor:

l¼

115

15.0

17.5

20.0

22.5

25.0

27.5

30.0

Vo2L2 nM0H Fig. 6. Comparison of the theoretical solution by Jones [40] (Eq. (4)) and the experimental results (Exp) of the members of length: (a) 600 mm, (b) 1000 mm.

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H.H. Jama et al. / Thin-Walled Structures 53 (2012) 109–122

The comparison of the global flexural deformation of the 600 mm and 1000 mm span experiments with Eqs. (5) and (8), not including and including strain rate effects, respectively, (using the values of D ¼884 and p¼2.207 as discussed previously), is shown in Fig. 6. The 300 mm length members are not compared to the theoretical solutions due to the prevalence of Mode II deformation, since tensile tearing is not included in the theory. The 600 mm members are mostly bound by the theoretical solution including strain-rate effects (Fig. 6a), while the theoretical solution performs poorly in comparison with the 1000 mm length members (Fig. 6b). The results in Fig. 6b are as expected, since the local deformation of the cross-section will clearly absorb a significant portion of the input blast energy. Therefore the experimental global flexural deformation should be lower than the theoretical solution, since there will be less energy available to deform the member flexuraly. Accordingly, the results in Fig. 6a are unexpected, since the theory performs relatively well. This is explained by the fact that the local deformations cause significant changes in the geometric properties of the

Photographs

cross-section, namely the second moment of inertia (I). The reduction of I results in significantly greater flexural deformation for the same applied load. Analysis of the reduction in I calculated from the measured deformed shapes revealed that for the more deformed shape of the ‘‘Tear drop’’, the final I value was as little as 30% of the original value [38]. In other words, the local cross-section deformation has two opposing effects on the global flexural deformation. On one hand, the local deformation absorbs energy which would indicate less energy being available for global flexural deformation. On the other hand, the local deformation decreases the flexural resistance of the member and therefore leads to increased global flexural deformation. Therefore, it is probable that the effect of the reduction in the flexural resistance of the 600 mm length members cancels out the effect of the energy absorbed in local deformation. This was less true for the 1000 mm members since the relative extent of cross-section deformation in these members was less (Table 1), due to the limitations on the explosive mass that could be used in the facility. Thus the use of a theoretical method that neglects local crosssection deformation is not generally applicable, and does not

Stationary Hinges Approach

Rolling Hinges Approach

Fig. 7. A sample of the 35  35  1.6 mm sections showing photographs and adopted mechanisms of the deformed cross-section at midspan using stationary or rolling hinges. Open circles (o) represent stationary hinges and dashed lines (–) represent the original cross section.

H.H. Jama et al. / Thin-Walled Structures 53 (2012) 109–122

provide accurate solutions for the flexural deformation. However, in the case where the members undergo very significant local cross-section deformations, such as occurs with the development of the ‘‘Tear drop’’ shape, then the theory may provide reasonable solutions for the flexural deformation. In the following section a theoretical method that is generally applicable to hollow sections of all sizes and under all magnitude of blast loads is developed.

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stretching or membrane forces developed as the crosssection changes shape and the circumference of the tube after the deformation is the same as prior to the deformation. (x) In the spirit of the approximate nature of the analysis and notwithstanding the fact there was variation in the measured yield stress of the members, the design static yield strength is taken to be the average of the measured values of 420 MPa.

5. Theoretical solutions for hollow sections

5.1. Assumptions (i) The first assumption is that the loading is impulsive, since the response times of the members are calculated to be between 1 ms and 2 ms based on the formulae of Jones [23], while the duration of the pressure pulse is estimated to be 0.04 ms based on the velocity of detonation of the explosive. According to Baker et al. [16] and the Health and Safety Executive (UK) [40], structures subjected to blast loads can be assumed to be impulsive provided the ratio of the pressure pulse duration to the natural period of the structure is less than 0.4. It is also significant to note that in the impulsive realm, the response is directly proportional to the impulse. (ii) Elastic deformations are ignored and the input impulsive load is assumed to be significantly greater than that which could be absorbed by the members in a wholly elastic manner. (iii) The local deformation of the cross-section is assumed to occur before the global flexural deformation. This is due to the inertial resistance of the member, and was confirmed with numerical solutions validated against the experimental results in [41]. (iv) Shear effects are ignored as the members have large length to depth ratios. The estimate of the errors involved in this assumption were quantified in [38] and shown to be less than 8.8%. (v) The total energy in the experiments is assumed to be consumed in the plastic deformation of the cross-section and plastic flexural deformation. Thus: I2 Etotal ¼ 2m

(vi)

(vii) (viii) (ix)

ð9Þ

where I is the measured impulse and m is the mass of the member under examination. Strain hardening is ignored in the analysis and is justified on the basis of the cold-formed steel being work hardened, resulting in a high ratio of the yield stress to the ultimate stress of 0.88. Strain rate hardening is accounted for using the Cowper– Symonds relationship. Rotary inertia is ignored but finite displacements are included in the analysis. The deformation in the circumferential direction is assumed to be in-extensional. That is, there is no circumferential

5.2. Energy consumed in local deformation using stationary hinge approach Stationary plastic hinge lines have been used by Zhao [42] and Zhao and Hancock [43] to determine the bearing capacity of hollow steel sections. Lu and Wang [44] also used the stationary plastic hinge approach to estimate the energy absorbed by square hollow tubes pierced by pointed punches under quasi-static loading conditions. It has also been used by Abramowicz and Jones [35,45] to estimate the energy consumed in the axial crushing of square and circular tubes. It is not the purpose of this paper to conduct an extensive review of the applications of yield line mechanisms and the interested reader is referred to the review by Zhao [46]. The determination of the energy consumed in the deformation of the cross-section begins with the simplification of the shape of the deformed cross-sections of the tubes. The deformed shapes of the cross-section at the midspan were imprinted onto ruled graph paper and stationary hinges and straight sections imposed on the shapes. For simplicity, the cross-sections were taken to be symmetrical. A sample of the simplified shapes is shown in Fig. 7 to illustrate the procedure. In general, as the impulsive loads are increased, the hinges increased from 4 to 14 in number. The deformed cross-sections were taken at the midspan of the members. After the determination of the final deformed crosssection and the simplification to a series of plastic hinges, the energy consumed in the local deformation can be calculated as

1.00

M/Mo

0.75 0.50 0.25 N/No

The rigid-plastic analysis of steel hollow sections subjected to transverse blast loads developed in this section consists of the analysis of the energy dissipated in the local cross-section deformations previously defined as the ‘‘Molar’’ and ‘‘Tear drop’’ shapes. Subsequently, the remaining applied energy is assumed to deform the member in global flexural deformation. Due to the complexity of the loading and response, the following assumptions are necessary to simplify the problem.

-1.00

-0.75

-0.50

-0.25

0.00 0.00

0.25

0.50

0.75

1.00

-0.25 -0.50 -0.75 -1.00 Fig. 8. Bending moment (M)—axial force (N) interaction yield curve for a square hollow section subjected to combined bending and axial force.

H.H. Jama et al. / Thin-Walled Structures 53 (2012) 109–122

Energy consumed in local deformation (CH) (Joules)

118

2500 35mm Beams 40mm Beams 50mm Beams

2250 2000 1750 1500 1250 1000 750 500 250 0 0

250

500 750 1000 1250 1500 1750 2000 Energy consumed in local deformation (RH) (Joules)

2250

2500

Fig. 9. Comparison of the energy consumed in local deformation for 600 mm length members, determined using the concentrated hinge (CH) and rolling hinge (RH) approaches.

Global midspan deformation (Delta/H)

2.50

20mm Exp 35mm Exp 40mm Exp 50mm Exp Bounds Average

2.25 2.00 1.75 1.50 1.25 1.00 0.75 0.50 0.25 0.00

5

0

10

20

15

25

Lamda (λf)

Global midspan deformation Detla/H [ - ]

3.0 2.5 2.0

35mm exp 40mm exp 50mm exp Bounds Average

1.5

1.0 0.5 0.0 0.0

2.5

5.0

7.5

10.0

12.5 15.0 17.5 Lamda (λf)

20.0

22.5

25.0

27.5

30.0

Fig. 10. Experimental midspan deformation (Exp) compared with the developed theory for hollow sections, for members of length: (a) 600 mm, (b) 1000 mm.

H.H. Jama et al. / Thin-Walled Structures 53 (2012) 109–122

along the member length was determined from experimental measurements.

follows: Elocal ¼

sy t 4

2

Lhinge

n X

119

y

ð10Þ

i¼1

where sy is the yield stress, t is the wall thickness of the tube, y is the angle of the mechanism and Lhinge is the length of the hinge lines along the member length. The most severe deformation occurred at the midspan and the least occurred near the supports, and the measured lengths of the mechanisms were between 0.5 and 1.0 times the member span. 5.3. Energy consumed in local deformation using rolling hinge approach Alternatively, the plastic energy consumed in forming the ‘‘Molar’’ and ‘‘Tear drop’’ shapes of the cross-section can be estimated based on rolling plastic hinges. A sample of the simplified shapes is shown in Fig. 7. The estimation of plastic work per unit length dissipated in yielding and complete flattening of tubular sections subjected to indentation was obtained by Reid and co-workers [47,48] and more recently by Elchalakani et al. [49]. Karagiozova et al. [50] used a similar approach to estimate the energy consumed in the deformation of metal hollow spheres. The energy consumed in the local deformation using the rolling hinge approach is determined as: ! Z n X 2M p y Elocal ¼ Lhinge rdy ð11Þ r 0 i¼1 i

where Lhinge is the length of the hinge lines along the member length, Mp ¼ syt2/4, y is the angle traversed by the rolling hinge and r is the radius of the rolling hinge. The rounded corners are included in the analysis. For simplicity, the cross-section is assumed to be symmetrical and generally three rolling hinges are formed. However, at higher impulses, an additional rolling hinge is formed at the bottom flange of the sections. Similar to the stationary hinge approach, the length of the rolling hinge

5.4. Energy consumed in flexural deformation and final midspan deformation Once the energy consumed in the local cross-sectional deformation is known, the global flexural deformation is estimated based on the principle of conservation of energy. That is, the energy dissipated in local deformation is subtracted from the applied energy and the remainder is assumed to be consumed in flexural deformation. The theoretical method developed by Jones [23] and discussed in Section 4 was used to determine the global flexural deformation. The bending moment–axial force interaction yield curve for a square hollow section is shown in Fig. 8 and is given by [38]:   M 4 N 2 ¼ 1 Mo 3 No

ð12Þ

where M is the plastic moment, Mo is the plastic moment capacity of the section, N is the axial force and No is the axial force capacity of the section. It is clear that the inscribing square yield curve intersects the exact curve at 0.5, which differs to the value of 0.618 obtained for solid sections by Jones [23]. The ‘‘lower bound’’ solution is furnished from the circumscribing square curve using sy, while for the ‘‘upper bound’’ solution sy is replaced with 0.5sy. An ‘‘average’’ solution was also calculated using 0.75sy. Strain rate hardening was included using Eqs. 6 and 8 with D and p of 844 and 2.207, respectively, as discussed previously. The foregoing solution is based on the premise that the local cross-sectional deformation precedes the global flexural deformation, thus the values used in Eq. (8) for H and Mp were those for the reduced cross-section. The plastic moment capacities of the deformed cross-sections (Mp) were calculated using the crosssection analysis software Thinwall [51], based on the measured post-test deformed shapes.

Table 2 Comparison of the experimental results and the semi-empirical design method. Test no.

Section bHt (mm)

Member length (mm)

Impulse (Ns)

Mo (Nm)

Mass/ metre (kg/m)

Slenderness ratio (le)

a

l

n

Global deformation experimental (delta/H)

Global deformation averagen (delta/H)

Experimental/ theoretical

32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53

35  35  1.6 35  35  1.6 35  35  1.6 35  35  1.6 35  35  1.6 40  40  1.6 40  40  1.6 40  40  1.6 40  40  1.6 50  50  1.6 50  50  1.6 50  50  1.6 50  50  1.6 50  50  1.6 35  35  1.6 35  35  1.6 40  40  1.6 40  40  1.6 40  40  1.6 50  50  1.6 50  50  1.6 50  50  1.6

600 600 600 600 600 600 600 600 600 600 600 600 600 600 1000 1000 1000 1000 1000 1000 1000 1000

39.1 49.2 56.0 58.3 66.8 39.0 49.1 61.4 64.7 47.8 67.1 72.6 76.2 94.1 107.7 125.4 90.1 107.7 125.4 90.1 107.7 125.4

1157 1157 1157 1157 1157 1535 1535 1535 1535 2457 2457 2457 2457 2457 1048 1095 1261 1261 1261 2101 2101 2099

1.63 1.63 1.63 1.63 1.63 1.88 1.88 1.88 1.88 2.38 2.38 2.38 2.38 2.38 1.63 1.63 1.88 1.88 1.88 2.38 2.38 2.38

28.0 28.0 28.0 28.0 28.0 32.2 32.2 32.2 32.2 40.6 40.6 40.6 40.6 40.6 27.8 28.4 30.5 30.5 30.5 39.3 39.3 39.3

0.79 0.79 0.79 0.79 0.79 0.90 0.90 0.90 0.90 1.11 1.11 1.11 1.11 1.11 0.30 0.30 0.36 0.36 0.36 0.58 0.58 0.58

5.1 8.0 10.4 11.3 14.8 3.3 5.2 8.2 9.1 2.0 3.9 4.5 5.0 7.6 135.0 175.0 59.4 85.0 115.0 22.5 32.2 43.7

1.3 1.4 1.5 1.5 1.6 1.2 1.3 1.4 1.5 1.2 1.3 1.4 1.4 1.5 2.6 2.9 2.0 2.3 2.6 2.3 2.7 2.7

0.52 0.78 0.92 0.79 0.97 0.30 0.45 0.71 0.64 0.41 0.56 0.64 0.85 1.05 1.07 1.13 0.67 0.89 1.25 0.64 0.85 1.05

0.55 0.73 0.84 0.87 1.00 0.45 0.60 0.78 0.82 0.36 0.56 0.62 0.65 0.83 1.63 1.78 1.26 1.45 1.62 0.84 0.97 1.17 Mean: COV:

0.95 1.07 1.10 0.90 0.97 0.66 0.74 0.91 0.77 1.13 0.99 1.03 1.29 1.27 0.65 0.64 0.54 0.62 0.77 0.76 0.87 0.89 0.90 0.19

n

The theoretical values obtained from Eqs. (14) and (15) as appropriate. Average refers to the average of the lower and upper bound solutions.

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5.5. Comparison of the theoretical solution with the experimental results The energy consumed in the local deformation determined using the stationary and rolling hinge approaches produce similar results as expected, and a comparison for a sample of specimens is shown in Fig. 9. There is some scatter in the results due to the approximate nature of the analysis. The analysis indicated that between 56% and 61% of the input energy was dissipated in local cross-section deformation. The experimental results are compared with the developed theory for hollow sections in Fig. 10a and b for the 600 mm and 1000 mm length members respectively. The energy consumed in local deformation was determined from the stationary hinge approach. The theory was not compared with the 300 mm members due to the prevalence of Mode II tensile tearing in these members (Table 1), which was not accounted for in the theory. It is clear in Fig. 10 that the experimental results are generally bounded by the theoretical solution, producing satisfactory results for the final midspan deformation of the members.

6. Design considerations When the energy consumed in local deformation and the resultant geometry changes are accounted for, the use of rigidplastic analysis furnishes bounded solutions for steel square hollow members subjected to transverse impulsive loads. Local cross-sectional deformations were found to be paramount in the analysis, however they produced two opposing effects. Local deformations absorb energy and thus lead to reduced global flexural deformation. Simultaneously, local deformations result in reduced cross-sectional properties and thus lead to increased global flexural deformation. Since the local cross-section deformations are complex and difficult to obtain apriori, a semiempirical solution is presented in this section that can be used to aid design. Therefore a new parameter (a) is introduced that reconciles the two opposing effects of the local cross-section deformation:

a¼

Ef lexure =Etotal ðEtotal Elocal Þ=Etotal ¼ Z reduced =Z local Z reduced =Z original

Global midspan deformation (Delta/H)

2.75 2.50

35mm Exp 40mm Exp 50mm Exp Bounds Average

2.25 2.00 1.75 1.50 1.25 1.00 0.75 0.50 0.25 0.00 0

3

5

8

10

13

15 

18

20

23

25

28

30

17.5

20.0

22.5

25.0

27.5

30.0

n

3.5

Midspan deformation Delta/H [-]

3.0 2.5

35mm Exp 40mm Exp 50mm Exp Bounds Average

2.0 1.5 1.0 0.5 0.0 0.0

2.5

5.0

7.5

10.0

12.5

15.0  n

Fig. 11. Experimental midspan deformation (Exp) compared with the semi-empirical design approach, for members of length: (a) 600 mm, (b) 1000 mm.

ð13Þ

H.H. Jama et al. / Thin-Walled Structures 53 (2012) 109–122

where Eflexure is the energy consumed in flexural deformation, Etotal is the input energy, and Zreduded and Zoriginal are the final and original flexural properties of the cross-section, respectively. The values of alpha (a) for the 600 mm and 1000 mm members are presented in Table 2. Recognising that the slenderness ratio (Eq. (1)) is the dominant non-dimensionalised parameter that determines local cross-sectional behaviour in thin-walled hollow sections, empirical relationships were then sought between the slenderness ratio and alpha (a), resulting in Eqs. (14) and (15) for the 600 mm and 1000 mm members, respectively:

a¼ a¼

le þ5 40

le 15 40

ð14Þ

ð15Þ

To estimate the final flexural deformation at the midspan of steel square hollow members, Eqs. (14) and (15) are used in conjunction with Eq. (4), resulting in Eq. (16):

d H

¼ 0:5ð

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ 0:75alÞ1Þ

ð16Þ

Eq. (16) is relatively simple, however considers the effects of finite displacements, strain rate hardening and geometry changes that have been found to be important in the analysis of SHS members subjected to impulsive loads. Eq. (16) was used in conjunction with Eq. (8), including strain rate effects (using the values of D¼ 884 and p ¼2.207 as discussed previously). The upper bounds were obtained as previously explained by replacing the yield stress by 0.5sy, and the average was obtained with 0.75sy. The simplified semi-empirical method is compared with the experimental results for the midspan deformation in Fig. 11 and Table 2. In Fig. 11 the method is shown to generally bound the experimental results, and in Table 2 the average solution is shown to compare reasonably well with a mean of 0.90. There is significant scatter in the comparisons due to the approximate nature of the analysis, resulting in a coefficient of variation of 0.19. However, given the complexities of the loading and the response, the solution is considered satisfactory.

7. Conclusions Experimental results of steel hollow sections subjected to transverse explosive loads have been shown to result in a combination of local cross-section deformation and global flexural deformation. More than 50% of the applied energy was found to be dissipated during local plastic deformation, while the remainder was dissipated in global plastic deformation. Existing rigid-plastic solutions for the global flexural deformation were developed for solid metal sections and thus neglect local crosssection deformations, and were found not to be generally applicable to steel hollow sections. A new theory was developed using rigid-plastic analysis and yield line mechanisms to determine the energy dissipated in local deformation, following which the remainder of the input energy was shown to provide bounded solutions for the flexural deformation. Using these results a semiempirical design method was developed, which provides reasonable estimates of the transverse flexural deformation of steel hollow section members subjected to transverse blast loads.

Acknowledgements The authors would like to thank the Department of Civil Engineering, Monash University and BISRU for the financial

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support, and acknowledge the Australian Research Council for providing a Discovery grant for this research. We would also like to thank Glen Newins, Horst Emrich and Peter Jacobs, of the University of Cape Town, for fabricating the test rig and Dr. Steeve Kim Chung Yuen (BISRU) for the helping with the experimental set-up. We would also like to thank Prof. Magnus Lanseth, Prof. Arild Clausen, Mr. Trond Auestad and Dr. Tharigopula Venkapathi of NNTU for assistance with the high strain rate testing. References [1] Bodner SR, Symonds PS. Experimental and theoretical investigation of plastic deformation of cantilever beams subjected to impulsive loading. Journal of Applied Mechanics—ASME 1962:719–28. [2] Humphreys JS. Plastic deformation of impulsively loaded straight clamped beams. ASME—Transactions—Journal of Applied Mechanics 1965;32(1): 7–10. [3] Symonds PS, Mentel TJ. Impulsive loading of plastic beams with axial constraints. Journal of the Mechanics and Physics of Solids 1958;6(3): 186–202. [4] Florence AL, Firth RD. Rigid-plastic beams under uniformly distributed impulses. ASME—Transactions—Journal of Applied Mechanics 1965;32(3): 481–8. [5] Menkes S, Opat H. Broken beams. Experimental Mechanics 1973;13(11): 480–6. [6] Bodner SR, Symonds PS. Experiments on dynamic plastic loading of frames. International Journal of Solids and Structures 1979;15(1):1–13. [7] Symonds PS, Chon CT. Large viscoplastic deflections of impulsively loaded plane frames. International Journal of Solids and Structures 1979;15(1):15–31. [8] Nurick GN, Jones N. Prediction of large inelastic deformations of T-beams subjected to uniform impulsive loads. San Francisco, CA, USA: ASME, New York, NY, USA; 1995. p. 127–53. [9] Wegener RB, Martin JB. Predictions of permanent deformation of impulsively loaded simply supported square tubes steel beams. International Journal of Mechanical Sciences 1985;27(1-2):55–69. [10] Bambach MR. Behaviour and design of aluminium hollow sections subjected to transverse blast loads. Thin-Walled Structures 2008;46(12):1370–81. [11] Nurick GN, Olson MD, Fagnan JR, Levin A. Deformation and tearing of blastloaded stiffened square plates. International Journal of Impact Engineering 1995;16(2):273–91. [12] Nurick GN, Gelman ME, Marshall NS. Tearing of blast loaded plates with clamped boundary conditions. International Journal of Impact Engineering 1996;18(7-8):803–27. [13] Langdon GS, Lemanski SL, Nurick GN, Simmons MC, Cantwell WJ, Schleyer GK. Behaviour of fibre-metal laminates subjected to localised blast loading: part I—experimental observations. International Journal of Impact Engineering 2007;34(7):1202–22. [14] Bambach MR, Zhao XL, Jama HH. Energy absorbing characteristics of aluminium beams strengthened with CFRP subjected to transverse blast load. International Journal of Impact Engineering 2010;37(1):37–49. [15] Baker WE. Modeling of large transient elastic and plastic deformations of structures subjected to blast loading. American Society of Mechanical Engineers—Transactions—Journal of Applied Mechanics Series E 1960;27(3): 521–7. [16] Baker WE, Cox PA, Westine PS, Kulesz JJ, Strehlow RA. Explosions Hazards and Evaluation. Elseveir Scientific Publishing Co; 1983. 273–364. [17] Louca LA, Boh JW, Choo YS. Design and analysis of stainless steel profiled blast barriers. Journal of Constructional Steel Research 2004;60(12): 1699–723. [18] Symonds PS, Chon CT. Bounds for finite deflections if impulsively loaded structures with time-dependent plastic behaviour. International Journal of Solids and Structures 1975;11(4):403–23. [19] Symonds PS, Chon CT. On Dynamic plastic mode-form solutions. Journal of the Mechanics and Physics of Solids 1978;26(1):21–35. [20] Jones N. A theoretical study of the dynamic plastic behavior of beams and plates with finite-deflections. International Journal of Solids and Structures 1971;7:1007–29. [21] Jones N, Gomes De Oliveira J. The influence of rotatory inertia and transverse shear on the dynamic plastic behavior of beams. Transactions of the ASME. Journal of Applied Mechanics 1979;46(2):303–10. [22] Symonds PS, Jones N. Impulsive loading of fully clamped beams with finite plastic deflections and strain-rate sensitivity. International Journal of Mechanical Sciences 1972;14:49–69. [23] Jones N. Plastic Failure of Ductile Beams Loaded Dynamically. ASME— Transactions—Journal of Engineering for Industry 1976:131–6. [24] Hodge JPG. Interaction curves for shear and bending of plastic beams. American Society of Mechanical Engineers—Transactions—Journal of Applied Mechanics 1957;24(3):453–6. [25] Karunes B, Onat ET. On effect of shear on plastic deformation of beams under transverse impact loading. American Society of Mechanical Engineers— Transactions—Journal of Applied Mechanics 1959:107–10.

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