Dynamic material performance of cold-formed steel hollow sections: a state-of-the-art review

Abstract

This paper presents a literature review focused on the material performance of cold-formed, carbon steel, hollow structural sections under impulsive (highly dynamic) loading. Impulsive loading, represented by impact and blast, is characterized by a very rapid, time-dependent loading regime in the affected members and materials. Thus, the effect of high-strain-rate loading is initially reviewed. Next the material toughness, an important energy-absorption property and one measure of a material’s ability to arrest fracture, is considered by means of studying the Charpy V-notch behavior. The response of hollow sections under axial and lateral impact loading is then reviewed. ‚‚Studies of blast on hollow sections, most of which fall under the categories of contact/near-field loading or far-field loading are presented. Under large-scale field blast experiments, cold-formed hollow sections have shown excellent behavior. Software for modeling blast loading and structural response, the latter including single degree of freedom analysis and explicit finite element analysis, is described and discussed.

Keywords

cold-formed steel / hollow structural sections / composites / impulsive loading / impact / blast / experimentation / analysis / material properties

Material properties under various strain rates

For impact- or blast-resistant design of steel and composite structures, it is important to use realistic material properties representative of their high strain rate behavior. In particular, the substantial rise in yield stress under high strain rate may have important effects on the dynamic performance of a structure. For example, the yield load and resistance of members may be raised, the cross-sectional classification of steel sections may be negatively impacted by this elevation of yield stress [], steel connectors (bolts and welds) may have increased strength but a corresponding reduction in ductility, members may have reduced deflection, and forces transferred into the structural framework may be different to those otherwise anticipated. For blast- and impact-resistant design, the most probable yield strength (already greater than the minimum specified yield strength, by a static strength increase factor) is multiplied by a dynamic increase factor (DIF_y≥1.0) to produce a dynamic yield strength for design. Similarly, a DIF_u multiplier (≥1.0) can be applied to the static ultimate tensile strength. The ultimate strength increase, however, is generally only slight, whereas the yield strength increase may be substantial. Compared to the static value, the modulus of elasticity remains nearly constant [, ].

Estimation of the applicable strain rate is difficult, with values potentially ranging from 0.1 s⁻¹ to 1000 s⁻¹ for a whole range of blasts and impacts. For example, it is estimated that the strain rates on the World Trade Center steels, due to aircraft impacts and close-proximity blasts, were up to 1000 s⁻¹ []. The DIF_y and DIF_u values for various structural steels, at relatively low strain rates, suggested by Gilsanz et al. [], are listed in Table 1. Scientific investigations have shown that steel with a higher static yield strength is less susceptible to the “strain rate effect,” and this is reflected in Table 1. Similar constant DIF_y and DIF_u values, independent of strain rate, are given in Refs. [–].

Based on the expected ductility ratio (ratio between the maximum displacement and the elastic displacement) or the expected support rotation angle (tangent angle at the support formed by the maximum beam deflection), it is suggested by AISC DG26 [] that the dynamic design stress for tension, compression and bending (f_ds) can be calculated as follows, for non-cold-formed steel, and again for low pressure explosions with strain rates in the order of 0.1 s⁻¹:

For ductility ratio≤10 or support rotation angle≤2 degrees,

(1){{\displaystyle f}}_{\mathrm{d}\mathrm{s}}={{\displaystyle f}}_{\mathrm{d}\mathrm{y}}+{{\displaystyle DIF}}_{\mathrm{y}}·{{\displaystyle f}}_{\mathrm{s}\mathrm{y}}.

where f_sy represents either the measured or most probable static yield stress of the steel.

For ductility ratio>10 or support rotation angle>2 degrees,

(2){{\displaystyle f}}_{\mathrm{d}\mathrm{s}}={{\displaystyle f}}_{\mathrm{d}\mathrm{y}}+\left({{\displaystyle f}}_{\mathrm{d}\mathrm{u}}-{{\displaystyle f}}_{\mathrm{d}\mathrm{y}}\right)/\mathrm{4},

where

(3){{\displaystyle f}}_{\mathrm{d}\mathrm{u}}={{\displaystyle {{\displaystyle DIF}}_{\mathrm{u}}·f}}_{\mathrm{s}\mathrm{u}}.

The dynamic design stress for shear is given as:

(4){{\displaystyle f}}_{\mathrm{d}\mathrm{v}}={{\displaystyle \mathrm{0.55}f}}_{\mathrm{d}\mathrm{s}}.

For higher-pressure explosions, where the loading is in close proximity, or for impact loading, UFC 3-340-02 [] recommends that the DIF_y for strain rates up to 100 s⁻¹ be determined using Fig. 1, which is given in terms of ASTM A36 and A514 plate steels (see f_sy values in Table 1).

The dynamic design stress for tension, compression and bending (f_ds), calculated using Eqs. (1) and (2), is illustrated in Fig. 2 for hot-finished (hot-formed) steel. As can be seen, Eq. (2) considers the strain hardening effect when the expected ductility ratio (i.e., the damage allowed in the structural member) is large.

Strain-rate effect on steel

The guidance documents reviewed in Section 1.1 are the result of many previous investigations into the “strain-rate effect” on steel, a number of which are summarized in Table 2. The dependence of the DIF_y on both the steel grade and the strain rate is repeatedly confirmed, with the DIF_y tending to increase logarithmically as shown in Fig. 3 for strain rates up to 1000 s⁻¹.

Research specifically on the effect of strain rate on the properties of hollow steel sections is very limited. For hot-finished (hot-formed) circular hollow sections (CHS) and rectangular hollow sections (RHS) one would expect that, since the properties of the hollow section are almost uniform around the cross section, the strain rate effect would mirror that of prior research on hot-finished steel. For cold-formed sections though, the degree of cold-forming around the cross section is likely to be influential, and particularly in the cold-formed corners of RHS. This effect was studied by Sun and Packer [], via a total of 166 split hopkinson pressure bar (SHPB) tests, performed in tension and compression and taken from the flat and corner regions of direct-formed and continuous-formed, cold-formed, RHS. They found that:

(a) RHS with a higher static yield stress are generally less susceptible to the strain rate effect;

(b) For direct-formed RHS the DIF_y does not change substantially as the cross-sectional geometry changes. For continuous-formed RHS the DIF_y increases as the RHS width-to-thickness ratio increases;

(c) For the same strain rate, the compressive and tensile DIF_y values for identical RHS may differ.

Representative average DIF_y values, for the entire cross-section, for RHS with different cross-sectional geometries and produced by different cold-forming methods were produced by Sun and Packer [], for use in the blast- or impact-resistant design of RHS members under compression, tension and flexural loadings. An example of these DIF_y tables is shown in Table 3. Unfortunately, because of the SHPB apparatus used, these DIF_y recommendations are confined to strain rates from 100 to 1000 s⁻¹, which confines their application to very close-in blast loadings or high impact.

Ritchie et al. [] have completed a series of high-strain-rate tensile coupon tests on cold-formed RHS using a high-speed actuator. These tests were initiated to produce data at lower strain-rates than that of Sun and Packer []. A total of 72 coupons was tested at strain rates from 0.1 to 20 s⁻¹. The preliminary results show good agreement with the work of Sun and Packer []. The authors plan on using this data to combine with others and calibrate material models in the future.

The combined effect of high strain rate loading and subsequent elevated temperature (e.g., due to fire after blast/impact) on mild steel has been recently investigated experimentally by Mirmomeni et al. []. Tests on coupons under uniaxial tension loading were performed in two consecutive phases. In Phase I, interrupted tensile tests – at two constant strain rates of 1 and 10 s⁻¹, at room temperature, were terminated at predefined displacements. In Phase II, carried out subsequent to Phase I, quasi-static tensile tests on these deformed coupons were carried out at elevated temperatures, up to 600°C. It was found that at temperatures above 450°C the temperature effect dominates, and that under extreme fire the strain rate (or pre-damage) does not have a consequential effect on the mechanical properties. Changes in the steel microstructure that take place under high-strain-rate loading and fire are further described by Mirmomeni et al. [].

When modeling the impulsive behavior of RHS members, the dynamic material properties need to be taken into account, for the anticipated peak strain rate associated with the type of loading. Single degree of freedom (SDOF) and explicit finite element (FE) programs will require dynamic increase factors, with either the Johnson-Cook [] or Cowper-Symonds [] material models being dominant in FE software.

Strain-rate effect on composite hollow sections

Filling a hollow section with concrete, or a cementitious grout, adds mass to the member and increases its energy-absorption properties, so this is often performed for blast or impact resistance. A considerable number of studies have been performed on concrete material under high strain rates, but it is important to note that the “size effect” phenomenon – known for concrete (static) test specimens – applies as well in the dynamic domain, albeit with a different (time-dependent) size effect [].

Much testing has been performed to determine the DIF for plain concrete in compression, resulting in the following empirical relationship given by the Comité Euro-International du Béton []:

(5){{\displaystyle DIF}}_{\mathrm{c}}={{\displaystyle \left[\mathrm{33333}\left(\dot{\epsilon }\right)\right]}}^{\mathrm{1.026}\alpha } \mathrm{f}\mathrm{o}\mathrm{r} \dot{\epsilon }\le {\mathrm{30}\mathrm{s}}^{-\mathrm{1}},

where

\alpha ={\left(\mathrm{5}+\mathrm{0.9}{{\displaystyle f}}_{\mathrm{c}}\right)}^{-\mathrm{1}}

and

(6){{\displaystyle DIF}}_{\mathrm{c}}=\gamma {{\displaystyle \left[\mathrm{33333}\left(\dot{\epsilon }\right)\right]}}^{\mathrm{1}/\mathrm{3}} \mathrm{f}\mathrm{o}\mathrm{r} \dot{\epsilon }>{\mathrm{30}\mathrm{s}}^{-\mathrm{1}},

where \gamma ={\mathrm{10}}^{\left(\mathrm{6.156}\alpha -\mathrm{2}\right)}.

The dynamic increase factor for concrete, DIF_c, is generally greater than that for steel but DIF_c also increases with increasing strain rate. Xiao et al. [] recently performed high-strain-rate compression tests on a modest number of concrete-filled circular tubes (CFTs) in a large SHPB apparatus. Comparative concrete cylinders were also tested under high strain rates similar to the CFTs and the CEB [] expression given by Eq. (6) predicted the DIF_c well. For the composite (CFT) specimens, the dynamic increase factor (DIF_CFT) was lower than that of plain concrete, at a particular strain rate. Thus: DIF_c>DIF_CFT>DIF_y. In this study the DIF_y was calculated from an expression by Malvar [] for steel reinforcing bars. It was then determined, although just from nine tests on small CFTs in compression and over a narrow strain rate range, that simple superposition of the axial strengths of the concrete infill and the steel tube – neglecting their interaction – gave a close, but conservative, under-estimate of the DIF_CFT. i.e.,

(7){{\displaystyle DIF}}_{\mathrm{C}\mathrm{F}\mathrm{T}}=\left({{\displaystyle DIF}}_{\mathrm{c}}{{\displaystyle f}}_{\mathrm{c}}{{\displaystyle A}}_{\mathrm{c}}+{{\displaystyle DIF}}_{\mathrm{y}}{{\displaystyle f}}_{\mathrm{s}\mathrm{y}}{{\displaystyle A}}_{\mathrm{s}}\right)/\left({{\displaystyle f}}_{\mathrm{c}}{{\displaystyle A}}_{\mathrm{c}}+{{\displaystyle f}}_{\mathrm{s}\mathrm{y}}{{\displaystyle A}}_{\mathrm{s}}\right).

Xiao et al. [] then proceeded to show that taking concrete confinement into account produced a more accurate prediction of the dynamic strength of CFTs.

Charpy V-notch impact toughness

The selection of steel for notch toughness is critical for low-temperature service or dynamic loading applications, due to the possibility of brittle fracture of a component. For the assessment of material toughness, international standards commonly require Charpy V-notch (CVN) impact testing of the steel product. A required toughness level is commonly expressed for a particular test temperature (which may be different to the lowest anticipated service temperature) at which a minimum CVN impact energy value (KV_min), shall be achieved. The approximate relationship between the CVN energy-temperature curve and the fracture behavior of a steel component is illustrated in Fig. 4.

Extensive investigations have been conducted on CVN impact toughness for a variety of cold-formed and hot-finished RHS and CHS []. A summary of the major sources used in Ref. [] is given in Table 4.

It should be noted that Ref. [] was later published as Ref. [] and Ref. [] was also published as Ref. []. The end result of Feldmann et al. [] is summed up in a simpler way by Sun and Packer [].

A review paper was written by Ritakallio and Björk [] on low-temperature ductility and structural behavior of cold-formed hollow section structures, mainly focused on high quality cold-formed Ruukki RHS, that even exceeds the manufacturing minimum requirements of EN10219 [,]. Another review paper by Puthli and Packer [] compared the structural performance of cold-formed hollow structural sections and hot-finished structural hollow sections. This paper listed all the arguments in discussion to show that, as long as the products are in accordance with European standards and a design is executed sensibly, cold-formed products perform efficiently in all structural applications. It is shown that cold-formed hollow sections which are correctly specified, correctly produced and supplied with the appropriate European test certification are suitable for all forms of construction. Sun and Packer [] showed that the cold-formed RHS in the rest of the world (i.e. which are not to EN10219) still have very mediocre and unreliable toughness, while hot-formed RHS always have excellent CVN properties.

Charpy V-notch impact toughness of cold-formed hollow structural sections

The toughness of cold-formed hollow sections depends not only on the toughness of the coil material used to manufacture the product, but also on the degree of cold-forming introduced to the cross-section during production. As illustrated in Fig. 5, cold-forming lowers the material toughness. In general, the cross-sectional geometry of the hollow section product is a good indicator of the degree of cold-forming contained in the section. For CHS, the toughness level around the cross-section is consistent since the coil material is cold-bent to the same curvature at all locations. On the other hand, for RHS, the toughness at the corner region can be significantly lower than that of the flat face due to uneven degrees of cold-forming, depending on whether the RHS was manufactured using the “direct-forming” or “continuous-forming” method. A comparison between material properties of North American direct-formed and continuous-formed RHS can be found in Sun and Packer [,].

The prime American standard for cold-formed hollow sections, ASTM A500 [], has no notch toughness requirement. Thus, it is necessary to specify CVN testing of the A500 product before using it for low-temperature service or dynamic loading applications. In another mass market, China, the national specification for cold-formed RHS used in building structures [] either requires no CVN value (Class II) or a modest CVN of 27 J at 20°C room temperature (Class I), although a lower temperature requirement can still be specified.

To offer cold-formed hollow sections suitable for dynamically loaded structures in North America, ASTM A1085 [] was developed recently. This specifies that, for hollow section product manufactured to this standard, its toughness shall be accessed by testing CVN specimens taken in the longitudinal direction (away from the seam weld) of the tube. The average CVN impact values of the test specimens shall conform to the minimum requirement of 34 J at 4°C, based on full-sized (10 × 10 mm with a 2 mm deep notch) test specimens. Such a CVN toughness level (at the test location) is adequate for dynamic loading application for the “Zone 2” service temperature range (−17.8°C to −34.4°C) as per the AASHTO bridge design specification []. However, it should be noted that, for RHS, ASTM A1085 [] – like all international standards – specifies that the CVN specimens be taken from the flat face of the tube. Unlike CHS, the toughness level around the cross-section of RHS is inconsistent due to non-uniform amounts of cold-forming. Thus, the CVN impact values of test specimens taken from the flat face do not necessarily represent the toughness property of the entire cross-section of the RHS.

Extensive investigations have been conducted on the effect of cold-forming on the toughness of European hollow sections, which formed the current rules for the selection of European hollow sections for overall notch toughness []. A survey of these investigations can be found in Sun and Packer []. However, since these tests were carried out mainly with hollow sections made of EN 10219 S355J2H steel [,], the rules in Ref. [] refer to this material type only (i.e., are not necessarily applicable to hollow sections produced elsewhere).

Similar investigations on North American hollow sections were limited until recently. Based on extensive CVN testing on hollow sections with different cross-sectional geometries and produced by different methods it has been concluded that [,]:

1) When selecting RHS for notch toughness, serious consideration should be given to the CVN toughness deterioration from flat face to corner (i.e., the weak spot) such that the entire cross-section is “fit for purpose”. This can be done by either specifying the corner as an alternate measuring location, or considering the deterioration from the flat face to the corner if the CVN toughness was measured in the standard location (flat face). Experimental results (see Fig. 6) showed that there are generally large temperature shifts (ΔT_cf) between the CVN energy-temperature curves of the flat face and the corner of the RHS tested. Such “temperature shifts” can be up to 20°C for continuous-formed RHS, or up to 40°C for direct-formed RHS, depending on the cross-sectional geometry of the RHS, and must be borne in mind by the designer. Thus, for example, a specification of 27 J at – 40°C in the flat region of a continuous-formed RHS would ensure a CVN rating of 27 J at – 20°C in the corner regions.

2) For CHS, since the toughness level is reasonably consistent around the cross-section, no “temperature shift” needs to be considered and uniform CVN toughness can be assumed.

3) For hollow sections with a wall thickness less than 11 mm, ASTM A370 [] specifies the use of sub-sized CVN specimens. Due to the fact that the width of the sub-sized specimen is reduced, it has to be notched on the narrow side (i.e., the specimen has a notch through the hollow section wall thickness) in order to have enough cross-sectional area for impact testing []. On the other hand, for thick-walled hollow sections where full-sized CVN specimens are possible, ASTM A370 permits the notch to be either on the hollow section surface or through the wall thickness. However, according to experimental evidence [], the through-thickness notch orientation generally produces a lower CVN toughness reading. This has also been confirmed by Ritakallio []. Hence, it is recommended that, for thick-walled hollow sections, full-sized CVN specimens should be machined with a through-thickness notch to produce conservative test results.

Impact behavior of hollow sections

Resistance of members under axial impact loading

The main application of hollow sections under axial impact can be found in bumper systems for vehicles. A typical bumper beam system consists of a bumper beam directly connected to a longitudinal at both ends, with the longitudinal members typically in the form of a RHS or CHS.

RHS under axial impact loading

Thin-walled steel RHS members, subjected to axial impact loading or axial quasi-static large-deformation loading, exhibit a multiple-fold collapse mechanism at failure, as shown in Fig. 7.

The load-deformation behavior (and hence energy absorption) of RHS members under impact load can be predicted by using plastic mechanism analysis or yield line theory. The energy method (also called the work method) is often adopted by assuming a rigid-plastic material and, using the condition of kinematic continuity on the boundaries between rigid and deformable zones, a basic folding mechanism is constructed. Two basic collapse elements were used by Abramowicz and Jones [] to simulate the folding mechanism of RHS as shown in Fig. 8.

Kohar et al. [] conducted FE analysis of RHS under a crushing force using LS-DYNA, wherein a nonlinear explicit dynamic formulation was used. Axial crushing simulations were performed on square tubes where the yield stress and strain, ultimate tensile strength, hardening rate, and failure strain of the material were varied.

A typical simulation is shown in Fig. 9. The main conclusions can be summarized as: (a) increasing the yield stress increases the energy absorption, peak crushing force and the steady-state crushing force; (b) the ultimate tensile strength, hardening rate and the yield stress together have a strong positive effect on the energy-absorption characteristics, while the failure strain has a weak contribution to the energy-absorption characteristics; (c) the peak crushing force is more sensitive to the yield stress than the steady-state crushing force; (d) increasing the yield stress produces an increase in crushing efficiency when the material hardening capabilities are higher.

CHS under axial impact loading

The bumper longitudinal in the shape of a CHS was studied by Pipkorn and Håland []. The failure mode of such circular hollow tubes was a progressive folding mechanism, which is similar to that [,] observed for CHS under quasi-static axial large-deformation loading. Plastic mechanism analysis was carried out [,] to predict the load-deformation curves of such tubes.

Resistance of members under lateral impact loading

The major applications of hollow sections under lateral impact loading can be found in bollard systems, building columns, roadside barriers and general-purpose energy absorbers. A summary of research work in this area is given in Table 5.

Bollards under lateral impact loading

Bollards are commonly used as barriers to prevent vehicles from obtaining access to a protected area, and CHS members are particularly common. The failure modes for steel CHS bollards under lateral impact load (Fig. 10) are very similar to those observed by Elchalakani et al. [] and Poonaya et al. [] for CHS under quasi-static bending. It should be pointed out that this kind of failure mode is valid for CHS which is either compact (with D/t<40) or non-compact (with D/t between 40 and 85).

Plastic mechanism analysis of a compact CHS has been performed by Elchalakani et al. [] to predict the moment-rotation relationship, and hence the energy absorption. The observed collapse mechanism can be represented by a yield line mechanism (YLM) as shown in Fig. 11. The same approach was adopted by Maduliat et al. [] for both compact and non-compact CHS under impact loading, plus they also conducted FE analyses using LS-DYNA on CHS bollards. Energy absorptions due to failure based on the YLM results and the FE results were compared, with the two methods giving very similar predictions. The energy absorption of the sections could, as expected, be increased by decreasing their diameter-to-thickness ratio. By changing the size of the bollards, their energy absorption due to global deformation did not change dramatically; however it had a significant effect on their energy absorption due to local deformation. The strength of thick-walled (stocky) sections is controlled by material yielding prior to (inelastic) local buckling.

Columns under lateral impact loading

Al-Thairy and Wang [] used numerical simulations to assess the static and dynamic approaches suggested by Eurocode 1 for designing steel columns to resist vehicle impact. The main conclusions from their study can be summarized as: (a) the equivalent static design force approach in Eurocode 1 is generally conservative for small- and moderately-sized columns that are typically used in low- and medium-rise buildings (less than 10 storeys); (b) if the column sizes are greater, using the Eurocode 1 equivalent static forces will over-estimate the axial compressive resistance of the columns, especially when the columns are used in structures located in rural areas or near national roads with high vehicle speeds. However, if vehicle velocities are low (<50 km/h), the Eurocode 1 equivalent static forces may still be used; (c) both the column and vehicle stiffness values should be included when calculating the equivalent impulse force–time relationship; (d) vehicle behavior should be divided into two stages – before the column is in contact with the vehicle engine, and after contact.

Al-Thairy and Wang [] developed a simplified analytical method to calculate the critical impact velocity of the vehicle that causes failure of the steel column. This method uses the energy balance principle and assumes a quasi-static response from the impacted steel column. The column is assumed to fail under global plastic buckling with a plastic hinge mechanism. The results from the simplified model agreed very well with those from a comprehensive set of numerical simulations using ABAQUS/Explicit. Although I-section columns were used as examples in the studies by Al-Thairy and Wang [,], the principle and methodology should be valid for hollow section columns.

Steel tubes are widely encountered in industrial applications (e.g., legs and bracing members of offshore oil platforms, building columns) and are commonly exposed to accidental loads. Prior to an accident, tubular structural members will be carrying their normal operational loads. Zeinoddini et al. [] showed experimentally that the effect of service load (or axial pre-loading) on the damage of tubular columns subjected to lateral impact is substantial.

Roadside barriers under lateral impact loading

Roadside barriers play an important role in minimizing the injury of vehicle passengers. Guardrails, such as the two-rail steel barrier and the three-rail steel barrier, are good examples. Wu et al. [] and Hao et al. [] have studied the performance of such roadside barriers under lateral impact loading, with FE simulations using LS-DYNA.

Energy absorbers under lateral impact loading

Tube-in-tube systems, using two and three nested tubes, have been studied by Wang et al. [] to investigate their energy absorption capacities. A typical three-tube system loaded in compression is shown in Fig. 12. A theoretical model, based on a rigid perfectly plastic material idealization, was established to predict the load-displacement response. It was found that the force-time response curves of these systems exhibit a “staircase” pattern, due to the progressive contacts between the tubes as they crush. Thus, desirable step-wise energy absorbers can be designed by selecting the geometry, material constants of each tube and the number of tubes.

Blast behavior of hollow sections

Blast loading, like impact loading, will produce much higher strain rates in the affected materials than non-impulsive dynamic load types (such as seismic or fatigue loading). The magnitude of blast loading is commonly represented by the scaled distance, Z(in units of m/kg^1/3). The response to blast loading will depend on the location of the blast source in relation to the responding element. Generally, blast loads can be classified as contact, near-field, or far-field. Several research projects have been undertaken to better understand the response of tubular steel to blast loading. Key parameters from these studies are listed in Table 6.

A comparison of the reference list to those listed during the discussion on lateral impact behavior illustrates that blast loading on tubular steel is a logical extension of research into impact loading. Much of the behavior seen with blast is very similar to impact, but there are some key differences that will be discussed.

Resistance of members under contact or near-field lateral blast loading

The cost and space required for large-scale blast testing make it difficult to perform. The space and charge size required to achieve significant results for contact or near-field blast trials is much less. This means it is feasible to repeat these tests and sometimes they can even be conducted in the secure confines of a laboratory. Repeatability and the use of a clean laboratory are desirable conditions for research. Therefore, the majority of the existing research into the blast behavior of hollow sections is on small-scale tubular specimens under contact or near-field loading.

One common apparatus used for small-scale blast testing is the ballistic pendulum. The ballistic pendulum uses conservation of energy and momentum principles to deduce the impulse applied by the explosive to the test specimen. This is important because the intense conditions produced near the explosive source make it difficult to measure the total impulse felt by the specimen in a different manner. Wegener and Martin [] were the first to use the ballistic pendulum on tubular steel. They measured the response of small (25×25 mm) RHS subject to sheet explosives distributed over the whole span. Their analysis of the local and global deformations, while simple, was effective at predicting the behavior of the tubular steel.

Extensive research using a ballistic pendulum and contact explosives was also completed by the same Monash University group that has studied the impact behavior of tubular steel. The Monash University blast research includes work on un-filled steel RHS [] and un-filled aluminum RHS []. A number of small-scale (25×25 mm to 50×50 mm) square RHS were tested with a high explosive distributed across the load face of the members. It was found that existing rigid-plastic solutions for solid cross-sections did not capture the local deformations found in the RHS and were not adequate for tubular steel. The authors proposed a new method that better captured the local deformation before using a rigid-plastic assumption. Simple design equations were developed based on these findings. Extensions of the experimentation and simple analytical modeling included explicit FE analysis of the response [] and measuring the effectiveness of carbon fiber reinforced polymer (CFRP) strengthening on the aluminum sections []. Figure 13 illustrates the experimental global and local deformations observed for a small steel RHS specimen, with increasing levels of impulse.

Bambach [] nicely summarizes the research group’s efforts and proposes generalized design methods, which are unified to include impact loading. The technique adds empirical modifications for the local deformations in the RHS to previously developed explicit solutions for solid sections. One issue with the technique is that it requires an impulse to be applied to the loaded face of the RHS. This can be achieved using the program ConWep [] for simple cases (as in the experimental work) but it requires complex numerical models for other cases that do not have a relatively uniform pressure.

Karagiozova et al. [] used the results of the Monash University experimental work [] to develop an analytical two-phase deformation model of RHS under near-field blast loading. During the first phase local collapse of the cross-section dominates the deformation and during the second phase the global deformation dominates. It was determined that, compared to solid sections, RHS dissipate a larger percentage of the total energy through plastic deformation. Karagiozova et al. [] expanded the analytical study and applied it to CHS. They found that, unlike RHS, for CHS local and global deformations happen simultaneously during the initial deformation phase. In the absence of experimental work on CHS, this study compared the analytical method developed previously to FE numerical results.

Other studies using near-field explosions have been completed using outdoor laboratory setups. Fujikura et al. [] studied a multi-column pier-bent with concrete-filled steel tube columns. A SDOF model was first used to design possible CFST configurations. A 1/4-scale model of the bridge was then constructed in the field and subjected to a typical vehicle bomb charge weight and standoff (also 1/4-scale). The authors found that the simple SDOF model was acceptable for the design of the CFST columns.

Remennikov and Uy [] conducted trials on single RHS members, both un-filled and concrete-filled, subjected to near-field blast loading. A total of five specimens were tested; four were concrete-filled RHS at varying standoff distances and one was an un-filled RHS used as a comparison. Figure 14 depicts the results of an un-filled and concrete-filled RHS subject to the same blast loading.

This research used a small charge centered over the specimen that led to an isolated local deformation pattern similar to the impact studies. A SDOF methodology was developed for predicting the response of the RHS members. This method converted the impulse applied by the blast loading into initial conditions on the RHS member. An FE model was also developed and good correlation was found for standoff distances above 0.15 m/kg^1/3. One difficulty with this experimental program, as with many near-field tests, is measuring the applied load or impulse. No such measurement was taken during this research and it relies on the assumption that the charge weight completely detonated.

Using the same experimental data, Ngo et al. [] completed a numerical analysis using an Arbitrary Lagrangian-Eulerian (ALE) method to apply the blast load. This analysis was able to capture both the local and global deformation of the RHS members. Like the Monash University work, it was found that there are two distinct phases to the RHS response to near-field blast loading: local deformations followed by global deformations. Concrete-filling was shown to serve as an effective energy-absorbing mechanism and the transverse deformation of concrete-filled specimens was much less than their unfilled counterparts.

Song et al. [] conducted a similar experiment on CHS, comparing experimental results to analytical and numerical models. Like previous studies discussed, the authors also found a good correlation between their experimental results and the numerical models.

Zhang et al. [] conducted tests on simply-supported RHS and CHS beams subject to near-field explosions. These tests were unique in that the beams were installed in a pit with only the load face exposed. This changes how the pressure wave interacts with the tubular member, as the ability of the wave to wrap around the member is greatly reduced. Figure 15 depicts a schematic of this setup, with an image of one of the RHS members installed.

Zhang et al. [] use an explicit FE numerical analysis to validate the blast tests. One drawback of their analysis is the use of a single pressure transducer (Fig. 15) to validate the use of the ConWep [] loading function for the numerical analysis. ConWep relies on several simplifications and it is hence not always applicable to near-field blast loads. This is an example of where the development of a computational fluid dynamics (CFD) numerical model to verify the use of ConWep should be done as a component of this research. As expected, concrete-filling reduces the local deformation in the steel and a higher percentage of the applied energy is absorbed by the global deformation.

Zhang et al. [–] used the same test setup as Fig. 15 to test RHS and CHS concrete-filled double-skin tube (CFDST) members subject to near-field explosions. The CFDST member had the annulus filled with with ultra-high-performance fiber-reinforced concrete (UHPFRC). In addition to a discussion of the test results [], Zhang et al. looked at the residual axial capacity of the members [] and carried out an explicit FE parametric study []. The authors found the response of the CFDST members to be similar to CFST tested [] and that the hollow core had little effect on the overall response. It was also found that when a small axial load was applied the measured peak lateral displacements were reduced.

Chen et al. [] used the same testing facility to test the post-fire blast response of reactive powder CFSTs. As expected, the maximum lateral displacement of the CFSTs increased with an increase in fire exposure.

Resistance of members under far-field lateral blast loading

The testing of specimens under far-field blast loading requires significantly more resources. Accordingly, there is much less research occurring in this area. One way of simulating far-field blast loading is to use a very large shock tube blast simulator. Such a facility exists at the White Sands Missile Range in New Mexico, USA, at which Clubley [] was able to test four CHS and RHS assemblies subjected to a simulated blast load. This research was primarily focused on the fluid dynamics of the blast wave as it passed by the small tubular steel members. The complex flow of the blast wave around members is an important consideration for analyzing structures comprised of many exposed tubular elements. The oil industry often has such structures and these are also at a high risk of being subjected to accidental or malicious explosions.

Another method of studying the behavior of members subject to far-field blast loading is to use blast arena testing. While still a very costly endeavor, blast arena testing allows a number of simultaneous experiments to be grouped around a single charge, thus improving the relative cost as more tests are added. This testing method is commonly used when the actual, real response to air-blast loading is required.

A group at the University of Toronto has been researching the behavior of large-scale, un-filled, concrete-filled, and concrete-filled double-skin tube (CFDST) hollow sections to far-field blast loading using blast arena testing [,]. These experiments have been performed on cold-formed RHS to EN10219 [,], with excellent member performance. Ritchie et al. have compared the un-fillled and concrete-filled results to SDOF models [] and explicit FE models [,]. Laboratory testing, including four-point bending tests, has been completed on the CFDST specimens and similar SDOF and FE comparisons are planned for the future [].

Due to the cost, numerical studies, with no experimental work, are common when studying the effects of far-field blast loads. Alternative methods of validation, such as a comparison to other numerical work or component-by-component validation, are often used. Zhai et al. [] conducted a numerical analysis of a large reticulated dome subjected to an explosive charge. Ding et al. [] completed a numerical analysis of the combined effect of blast loading and fire on tubular elements. This study used a two-part analysis to examine these loads. First, blast loading was applied. Secondly, a temperature load was applied to the deformed member to simulate fire damage and measure the stability of the element. This study resulted in pressure-impulse (P-I) damage diagrams, which are a valuable tool used in the design of protective structures. With P-I diagrams the designer needs to determine a likely pressure and impulse to which an element will be subjected (with assumptions as necessary), and then the damage can be assessed from the diagram.

Numerical and analytical modeling of blast

A common theme through many of the blast studies is numerical analysis, whether using SDOF or explicit FE methods. The high cost and difficulty of executing repetitive experimental work in this area make numerical analysis an extremely valuable, and often necessary, tool.

Single degree of Freedom

SDOF analyses are popular for design and analysis of structural elements subject to blast loads. As the research cited previously shows, this extends to tubular members. By far the most common SDOF tool used in blast design is the Single-degree-of-freedom Blast Effects Design Spreadsheet (SBEDS) developed by the US Army Corps of Engineers []. Researchers and designers concerned with hollow sections will almost exclusively use the “Steel Beam or Beam-Column” input sheet from SBEDS. Within this input sheet, users can input section properties, material properties, and various static and dynamic loads. Blast pressures can also be input manually, from a text file, or using the ConWep load tables (discussed below). The flexibility provided by SBEDS enables it to be used for a variety of problems.

ConWep [] determines various blast wave parameters, including pressure, impulse, and time of arrival, based on charge weight and standoff. The program refers to so-called “spaghetti charts” (Fig. 16) developed by the US Department of Defense [] that have proven to be an accurate representation of blast waves for many typical cases. However, caution must be used when the blast loading that is to be represented is too far from the typical uniform cases used to develop the “spaghetti charts”. This is true for many of the contact/near-field projects cited previously.

Finite element analysis

For more complex load cases, or when more accurate results are required, nonlinear explicit finite element analyses are used. Many of the research projects cited earlier in this section have completed such analyses. The most commonly used finite element software is the explicit program, LS-DYNA []. ABAQUS/Explicit [] and ANSYS Autodyn [] also have been used for blast loading on structural elements. LS-DYNA is the preferred software for the options it provides with respect to blast loading on structural elements. Built into the code is a function that will apply ConWep [] loads to an element. LS-DYNA also has computational fluid dynamics (CFD) capability with the arbitrary Lagrangian-Eulerian (ALE) method. This allows users to tailor their model to the complexity of their problem. When the ConWep loading will provide a good approximation of the load, such as for far-field blast loading on a façade, it can be used. However, when ConWep will not suffice the much more computationally expensive ALE solver can be used. For blast pressures on individual exposed hollow section members, where fluid flow around the member takes place, CFD analysis will be necessary, even for far-field blast loading. Both “prescribed” loading and “CFD-based” loading have a place in research on blast behavior of hollow sections.

Another consideration for modeling the blast response of hollow sections is the material dynamic increase factor (DIF) to be incorporated, as discussed in Section 1. Models that cater for the strain-rate effect, and which are commonly used for steel, are given by Johnson and Cook [] and by Cowper and Symonds []. These models both give DIF_y expressions which have empirically justified constants, so care must be taken to ensure that they are applicable for the steel type and strain rate under consideration.

Conclusions

The utilization of cold-formed steel hollow sections in structures governed by quasi-static, fatigue and seismic loading is now wide-spread. Knowledge of the material properties pertinent for such design situations is readily available and design engineers are generally aware of the performance of steel hollow sections under such load cases. Now, however, cold-formed steel hollow sections are being applied in designs involving highly dynamic loading rates, associated with impact or blast loading. This paper thus reviews research on the performance of cold-formed steel hollow sections under high loading rates, and summarizes material characteristics, member behavior and analysis methods/tools for designers. An extensive bibliography is provided as a basis for future research and development in this field.

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