Influence of elevated strain rate on the mechanical properties of hollow structural sections

Abstract

As protective design engineering becomes more prevalent, cold-formed steel hollow structural sections are often desired design components. As such, it is necessary to understand the behavior of hollow structural sections subject to air-blast loading, including the material response under elevated strain rates. Dynamic tensile tests have hence been performed on subsize tensile coupons taken from the flats and corners of cold-formed rectangular hollow section members. Dynamic yield stresses were obtained at strain rates from 0.1 to 18 s⁻¹, which encompasses and exceeds the range recorded during far-field blast arena testing. The dynamic increase factor was calculated for each data point and synthesized with previous cold-formed rectangular hollow section tests at even higher strain rates (100–1000 s⁻¹). The data set was used to determine Cowper–Symonds and Johnson–Cook parameters. The resulting material models can now be used to determine the strength increase of cold-formed rectangular hollow sections subject to a wide range of impulsive, elevated strain rate loads.

Keywords

Hollow structural section rectangular hollow section cold-formed steel elevated strain rate blast impact dynamic increase factor impulsive loading

Introduction

It has been established that steel is a strain-rate-sensitive material; it exhibits different characteristics, namely, an increased material strength with increasing strain rates. Dynamic loads, such as blast or impact, can produce intermediate (0.1–100 s⁻¹) to high (>100 s⁻¹) strain rates that significantly alter the material behavior. Around the world, the frequency of blast loading events, both accidental and malicious, is increasing. Therefore, modern designers recognize the value of incorporating protective design measures into even civilian infrastructure. With greater attention given to the protective design aspects of all structures, understanding the behavior of steel under elevated strain rates is becoming increasingly important.

Dynamic strength increases are critical for calibrating the numerical models that are often employed in protective analysis/design. Whether a simplified single-degree-of-freedom (SDOF) model or a complex finite element (FE) model is chosen, the material behavior at elevated strain rates is required. Previous research on strain rate behavior of steel subject to blast loading has identified strain rates on the order of 100 to 1000 s⁻¹ (Ngo et al., 2007). Accordingly, the majority of the recent research into the strain rate behavior of structural steel has been conducted in this range (Cadoni and Forni, 2015; Forni et al., 2016; Luecke et al., 2005; Marais et al., 2004; Sun and Packer, 2014). However, strain rates of that magnitude are not present in all blast events. Recent air-blast tests on cold-formed rectangular hollow section (RHS) specimens have resulted in intermediate strain rates on the order of 1 s⁻¹. This agrees with what others have found recently during air-blast loading of full-size steel W-sections (Nassr et al., 2014). Therefore, there is a need to establish material models, for use in protective design, that cover the whole spectrum of possible strain rates.

Hollow structural sections (HSS) are often used in commercial, institutional, and residential construction, especially when architecturally exposed elements are desired. As protective design becomes more mainstream, there is a desire to utilize traditional structural systems, rather than the massive concrete systems often used in protective design. Therefore, there is a need for accurate material models for HSS steel. This is especially true for cold-formed HSS steel, the type employed in North America and most of the world, as it has several unique characteristics when compared with other hot-rolled steel elements. These features are due to the residual stresses in cold-formed HSS that are a product of the manufacturing process.

Previous investigations

Elevated strain rate testing

One of the reasons for the lack of elevated strain rate material testing on the order of 1 s⁻¹ is the difficulty of obtaining the desired strain rates. Previous investigations typically involve one of two-test apparatuses: a tensile universal testing machine or a Split-Hopkinson Pressure Bar (SHPB) apparatus. Universal testing machines have been used to cover the lower values of the strain rate spectrum, typically from static tests (~1 × 10⁻⁵ s⁻¹) through to the upper limit of earthquake strain rates (~1 s⁻¹). As advancements have been made in universal testing machines, such as faster servo-hydraulic valves, the upper bound of their capabilities has increased. At the upper end of the spectrum, the lowest strain rates that the SHPB apparatus is capable of achieving are approximately 100 s⁻¹. Other impact-based test setups have exhibited similar strain rate ranges. This lower limit is a function of the impact mass used to generate the test impulse, where impulse is the product of moving mass and velocity change. Generation of a sufficient test impulse at low velocities requires an impractically large impact mass. Table 1 lists the details of previous research into the strain rate behavior of structural steels.

Table 1.

Strain rate test setups and ranges of previous research.

Resource	Test setup	Dynamic strain rate range (s⁻¹)	Material	Reported f_y (MPa)	No. of specimens
Manjonie (1944)	Screw-power-type testing machine	8.5 × 10⁻⁴–5.0 × 10⁻¹	Low-carbon steel	~200	2
	High-speed impact	100–1000			5
Rao et al. (1966)	Screw-power-type testing machine	1.1 × 10⁻⁵–1.4 × 10⁻³	Various structural steels	238–778	257
Campbell and Cooper (1966)	Servo-hydraulic testing machine	1.0 × 10⁻³–106	Low-carbon mild steel	200–300	15
Yu and Jones (1991)	Servo-hydraulic testing machine	1.2 × 10⁻³–140	Mild steel	~250	15
Kassar and Yu (1992)	Servo-hydraulic testing machine	1.0 × 10⁻⁴–1	Sheet steels	210–610	178
Filiatrault and Holleran (2001)	Servo-hydraulic testing machine	5.0 × 10⁻³–1.0 × 10⁻¹	Reinforcing bar	400	36
Marais et al. (2004)	SHPB	360–918	Mild steel	215	5
Luecke et al. (2005)	Servo-hydraulic testing machine	50–500	Structural steel columns	248–621	136
	SHPB	389–2899			17
Sun and Packer (2014)	SHPB	100–1028	HSS steel	350	166
Mirmomeni et al. (2015)	Servo-hydraulic testing machine	1–10	Hot-rolled mild steel structural plate steel	350	84+
Cadoni and Forni (2015)	Hydro-pneumatic machine	5–25	Reinforcing steel	500	6
	SHPB	250–1000			9
Forni et al. (2016)	Hydro-pneumatic machine	5–25	Structural steel, S355	355	6
	SHPB	300–850			9

MPa: megapascals; SHPB: Split-Hopkinson pressure bar; HSS: hollow structural sections.

The tests listed in Table 1 cover a wide range of strain rates and structural steel types. While there are a few tests that cover strain rates from 1 to 10 s⁻¹, most of them, with the exception of Mirmomeni et al. (2015), have only a small number of specimens tested. In addition, there is only one test program, Sun and Packer (2014), that uses cold-formed HSS steel. Another important factor is the yield strength of the tested material. It has been shown that the relationship between dynamic increase factor (DIF) and strain rate is dependent on yield strength. Generally, lower yield strength steel exhibits a higher DIF (Mainstone, 1975). Many of the oldest tests were on steel manufactured to historical standards, which have a much lower yield strength than modern day steel, as identified in Table 1. This illustrates the need to implement a test procedure that allows for a significant number of cold-formed specimens tested in the desired strain rate range. This is the only way to develop an up-to-date and comprehensive relationship between material behavior and strain rate for modern cold-formed RHS.

Elevated strain rate material characterization

Strain rate investigations are needed to calibrate strain-rate-dependent material models, which can then be used for protective design. The Department of Defense’s (DOD) Unified Facilities Criteria (UFC) 3-340-02 (DOD, 2008), the essential protective design guide, notes that under increased strain rate only the yield and ultimate stress change significantly and that the modulus of elasticity and the rupture strain of steel are generally insensitive to strain rate. Therefore, protective design typically uses a DIF, the ratio of the dynamic to static stress, to define the material behavior under elevated rates of loading. Material models used for protective design range from simple strain rate independent DIF values to more complex formulations that vary with strain rate. The CSA S850-12 (Canadian Standards Association, 2012), Canada’s design standard for blast loads, lists three options for determining the DIF of a material: testing or refined methods of analysis, a table of constants for various materials, or simple functions of strain rate.

For simple analyses or SDOF modeling, constants are generally accepted. However, for more complex analyses, such as FE analysis, material models that consider strain rate are desired. Over the years, a number of models have been proposed. An example of a strain rate dependent model, adapted from UFC 3-340-02 (DOD, 2008), is shown in Figure 1.

Figure 1.

DIF_y versus strain rate for A36 and A514 steel (adapted from Department of Defense [DOD], 2008).

One of the difficulties with calibrating and using the more complex material models based on strain rate is the differing test methodologies. As illustrated in Table 1, a variety of test setups have been used for steel material testing under elevated strain rates. The majority of these setups also employ unique instrumentation plans. Not only is the method for obtaining the data different, but the data analysis methods are different as well. There is not a universally accepted method for calculating the strain rate for a given test. Some of the methods used have included a linear fit of strain rate to yield (DOD, 2008), a linear fit of strain rate to a stage between yield and ultimate (Cadoni and Forni, 2015; Forni et al., 2016), and the absolute maximum strain rate recorded during the test (Nassr et al., 2014). Without a commonly accepted standard for strain rate determination, two researchers could take the same data set and reach different conclusions.

Nonetheless, several complex material models have been proposed for use in protective design and analyses. For example, LS-DYNA (Livermore Software Technology Corporation [LSTC], 2014), a commonly used explicit FE software, has several built-in methods of incorporating strain rate effects. These methods vary from manually inputted stress-strain curves to constitutive equations. For steel, typical methods include linear piece-wise stress-strain curves, the Cowper–Symonds model (Cowper and Symonds, 1957), the Johnson–Cook model (Johnson and Cook, 1983), or the Zerilli–Armstrong model (Zerilli and Armstrong, 1987).

The piece-wise model incorporates several stress–strain curves at different strain rates. A linear interpolation is used when the strain rates measure between tabulated curves. When the strain rate is below the lowest curve, the lowest curve values are used. Similarly, the highest strain rate curve is used if the strain rate falls at or above the highest strain rate. Using this method requires data for the given material at several strain rates. While this is possible when conducting a detailed analysis, it is not conducive to design.

In 1957, Cowper and Symonds developed a constitutive model that defines DIF as a function of strain rate using a power–law relationship. It is shown in Equation (1)

D I F = \frac{f_{d y}}{f_{y}} = 1 + {(\frac{\overset{\cdot}{ε}}{C})}^{\frac{1}{q}}

(1)

where C and q are Cowper–Symonds fit parameters.

In 1983, Johnson and Cook created a model that captures the strain rate and temperature dependence of a material. The Johnson–Cook model is usually referred to by its three terms: the strain-hardening behavior, the logarithmic strain rate sensitivity, and the temperature dependence. The model is shown in Equation (2)

f = (a + b ε_{p}^{n}) (1 + c \ln ({\overset{\cdot}{ε}}^{*})) (1 - T^{* m})

(2)

where a, b, c, n, and m are Johnson–Cook parameters, ${\overset{\cdot}{ε}}^{*} = \overset{\cdot}{ε} / {\overset{\cdot}{ε}}_{0}$ is the dimensionless strain rate, and T^* is defined by Equation (3)

T^{*} = \frac{T - T_{r o o m}}{T_{m e l t} - T_{r o o m}}

(3)

where T_room is a reference temperature and T_melt is a reference melt temperature.

In 1987, Zerilli and Armstrong developed a constitutive equation in an attempt to better describe material behavior. Their model is based on simplified dislocation mechanics and has face-centered cubic (fcc) and body-centered cubic (bcc) forms. The bcc version of the model is shown in Equation (4)

f = c_{0} + c_{1} e^{(- c_{3} T + c_{4} T \ln (\overset{\cdot}{ε}))} + c_{5} ε^{n}

(4)

where c₀, c₁, c₃, c₄, c₅, and n are Zerilli–Armstrong parameters.

The complexity of the material model chosen depends on the application. Due to its simplicity, the Cowper–Symonds material model is often used in FE analyses to quantify the strain rate dependence of structural steel members subject to blast loading (e.g., Krishnappa et al., 2013; Ngo et al., 2015; Zhang et al., 2015). When considering far-field air-blast loading the influence of temperature is lessened. The simplicity of the Cowper–Symonds model is also well suited to design, where the exact details of the material are often unknown. For these reasons, with the application to the aforementioned RHS tests in mind, the Cowper–Symonds model was chosen for further detailed analysis.

Cowper–Symonds material model

Since the model was originally proposed by Cowper and Symonds in 1957 for mild steel, numerous researchers have proposed new Cowper–Symonds parameters for various materials. Several researchers have used this material model for structural steel, including a number of those identified in Table 1. Table 2 provides a sample of various structural steel investigations using the Cowper–Symonds model and the resulting parameters.

Table 2.

Cowper–Symonds parameters from previous investigations.

Resource	Strain rate range (s⁻¹)	Reported f_y (MPa)	C (s⁻¹)	q
Cowper and Symonds (1957)	9.5 × 10⁻⁷–300	~200	40	5.000
Abramowicz and Jones (1986)	40–195	273	6844	3.910
Yu and Jones (1991)	1.2 × 10⁻³–140	~250	10,500,000	8.300
Marais et al. (2004)	360–918	215	844	2.207
Jama et al. (2009)	83–403	430	844	2.207
Sun and Packer (2014), tensile flats	191–997	350	2729	2.443
Sun and Packer (2014), tensile corners	156–1028	350	3280	1.292
Cadoni and Forni (2015), B500A	5–1000	564	141,917	2.816
Cadoni and Forni (2015), B500B	5–1000	571	599	0.753
Forni et al. (2016)	5–850	448	4945	2.696

MPa: megapascals.

The graphical representation of the Table 2 resources is shown in Figure 2. The Cowper–Symonds curves for all resources have been extrapolated across the full dynamic strain rate range, even though they may not be empirically valid. This is in line with the objective of using a single set of parameters for an analytical model, regardless of the actual strain rate range covered.

Figure 2.

DIF_y versus strain rate for various Cowper–Symonds model parameters.

Comparison of the graphical representations leads to the conclusion that there is a large variation in the formulation of DIF curves, with more contemporary research deriving much lower DIF_y values than those originally proposed by Cowper and Symonds (1957). This extreme variation highlights the need for Cowper–Symonds parameters specific to the material under consideration. Use of inaccurate DIF formulations can have a significant effect on FE results.

Johnson–Cook material model

One of the reasons the Johnson–Cook model is often used is the ease with which it can be calibrated for new materials. The three terms that make up the material model are easily isolated. The strain rate effects can be defined as a DIF, as shown in Equation (5)

D I F = 1 + c \ln ({\overset{\cdot}{ε}}^{*})

(5)

where c is the Johnson–Cook strain rate parameter and ${\overset{\cdot}{ε}}^{*} = \overset{\cdot}{ε} / {\overset{\cdot}{ε}}_{0}$ is the dimensionless strain rate.

Since the model was developed by Johnson and Cook in 1983, several researchers have developed parameters for a variety of new materials. While it has been not been cited as often as the Cowper–Symonds model for structural steels, select Johnson–Cook investigations are detailed in Table 3. A graphical representation of these investigations is shown in Figure 3.

Table 3.

Johnson–Cook parameters from previous investigations.

Resource	Strain rate range (s⁻¹)	Reported f_y (MPa)	${\overset{\cdot}{ε}}_{0}$	c
Johnson and Cook (1983), 1006 steel	2.0 × 10⁻³–600	350	1.0 × 10⁻¹	0.022
Børvik et al. (1999)	1.0 × 10⁻⁴–1000	490	5.0 × 10⁻⁴	0.012
Seidt (2005)	1.0 × 10⁻⁴–3000	286	1.0 × 10⁻⁰	0.017
Cadoni and Forni (2015), B500A	5–1000	564	Not reported	0.01769
Cadoni and Forni (2015), B500B	5–1000	571	Not reported	0.02139
Forni et al. (2016)	5–850	448	1.0 × 10⁻¹	0.0247

MPa: megapascals.

Figure 3.

DIF_y versus strain rate for various Johnson–Cook model parameters.

A key feature evident from Figure 3 is that the Johnson–Cook DIF function is highly dependent on the user-defined reference strain rate ( ${\overset{\cdot}{ε}}_{0}$ ). Not only does this value strongly influence the DIF curve, but it is also needed to use the model in the future. When it is not included, such as in Cadoni and Forni (2015), it renders the reported c value unusable for future modeling.

Field blast testing

In 2012, 2013, and 2015, large-scale blast arena testing was conducted by the authors on RHS elements. A blast arena test configuration was used to test the flexural behavior of the RHS elements subject to far-field air-blast loading. The 2012 test series examined unfilled RHS of two sizes (RHS 150 × 150 × 10 and RHS 150 × 150 × 8) and two different scaled distances (2.5 and 2.0 m/kg^1/3). In 2013, unfilled and concrete-filled RHS were tested side by side to facilitate a comparison of their behavior. Again, two RHS sizes were tested (RHS 120 × 120 × 8 and RHS 120 × 120 × 5) at two different scaled distances (2.7 and 1.9 m/kg^1/3). In 2015, two concrete-filled double-skin tube (CFDST) configurations, using an inner (RHS 80 × 80 × 3 or RHS 60 × 60 × 3) and outer (RHS 120 × 120 × 6) RHS element with the annulus between filled with concrete, were tested at two scaled distances (1.8 and 1.7 m/kg^1/3). The hollow section elements were loaded into a concrete reaction structure and steel cladding was used to load them in identical pairs. Figure 4(a) illustrates the 2015 hollow section target just prior to testing. A computer model depicting the hollow section reaction structure is shown in Figure 4(b), showing the elements with and without cladding. For all three-test series, the hollow section elements were instrumented at midspan with a horizontal displacement gauge to measure peak displacement and two strain gauges measuring the longitudinal strain at midpoint. Figure 4(c) shows the midspan mounting location on the hollow section elements during the 2015 test series. The RHS members in all tests failed in flexure while experiencing large inelastic deformations without fracture of the steel material, which is the traditional objective in blast-resistant design.

Figure 4.

2015 field test series: (a) hollow section target, (b) computer model of hollow section target, and (c) strain gauges and displacement gauge affixed at the midspan of hollow section member.

The strain gauges were used to get an accurate measurement of the strain rate in the RHS elements during the blast loading. For steel elements, the strain rate used for design and analysis is often measured as the linear strain up to the yield strain value (DOD, 2008). When possible, this method was used for the determination of the strain rates from the blast arena tests. Due to the extremely high forces involved with the blast loading, the strain gauges often did not stay adhered for the whole duration of the RHS oscillations and occasionally failed before yield. In the event that a strain gauge failed before yield strain, a linear fit of the data until failure was used to determine the strain rate. Typical strain versus time histories for the two tests (T1 and T2) from the 2015 test series are shown in Figure 5. The time scale of both curves has been normalized to the time of arrival of the blast wave at the target.

Figure 5.

Typical measured strain versus time with linear strain rate fit.

Strain rates to yield from the 2015 tests and 2013 tests on unfilled and concrete-filled RHS are shown in Table 4.

Table 4.

Strain rates measured during the 2015 and 2013 test series.

Year	Specimen	Strain rate (s⁻¹)
2015	15-T1-6DS60-1-R^*	1.05
	15-T1-6DS60-2-R	1.32
	15-T1-6DS80-1-R	0.98
	15-T1-6DS80-2-L	1.28
	15-T2-6DS60-2-L	2.23
	15-T2-6DS60-2-R	2.42
2013	13-T1-5H-1-R	0.50
	13-T1-8H-1-R	1.41
	13-T1-5CF-1-R	1.13
	13-T1-8CF-1-R	0.59
	13-T2-5H-1-R	3.33
	13-T2-8H-1-R	0.97
	13-T2-5CF-2-R	1.60
	13-T2-8CF-1-R	1.19

Legend: Year-test number-specimen type-specimen number-strain gauge location.

The values measured in 2015 are very similar to those recorded in 2013, which used a comparable loading. These values illustrate the need for material models that incorporate intermediate strain rate values on the order of 1 s⁻¹. These recorded strain rates are also very similar to those recorded by Nassr et al. (2014) whose air-blast loading of steel W-sections resulted in strain rates from 0.2 to 5.0 s⁻¹.

Laboratory testing

During the manufacturing of the 2015 field blast test specimens, extra RHS lengths of the identical (untested) material were retained and shipped to the laboratory for further material testing.

HSS specimens

The cold-formed RHS elements used for the 2015 test series were manufactured to EN10219 Grade S420MH and S355J2H (dual-graded; European Committee for Standardization [CEN], 2006). Three hollow section sizes were used: RHS 120 × 120 × 6, RHS 80 × 80 × 3, and RHS 60 × 60 × 3. First, full-size standard sheet coupons (ASTM International, 2015) were cut out of one end of each RHS length for static tests. The remainder of the members were divided into 18 segments (“bands”) of 111 mm length to manufacture subsize sheet coupons (ASTM International, 2015) for use in the high-speed actuator tests.

The subsize coupon geometry was selected for the intermediate strain rate coupons because it has a shorter gauge length, allowing for theoretically higher strain rates to be reached, for a given displacement rate during testing. Displacement rate is a crucial parameter because the maximum strain rate achieved using a high-speed actuator is governed by the maximum displacement rate that the actuator can reach.

Six full-size static coupons were cut out of each RHS size, three from the flat faces (excluding the weld seam face) and three from the corners. For the intermediate strain rate coupons, each band was labeled with seven possible coupon locations (A-G): three flat faces (excluding the weld seam face) and four corner locations (Figure 6a). Four subsize coupons (Figure 6b) were then machined from each band, two from the flat regions perpendicular to the weld seam (B & F), one from the corner with the largest measured radius (varies) and one from the corner with the smallest radius (varies). The intermediate strain rate coupons were appropriately labeled (see Figure 6c) for record-keeping purposes.

Figure 6.

Intermediate strain rate test coupons, (a) location around the RHS, (b) machined shape, and (c) naming convention.

Static tensile coupon tests

The standard full-size tensile coupons were tested in accordance with ASTM A370-15 using a 1000 kN-capacity MTS universal testing machine (MTS245; Figure 7a). Load was measured using the built-in load cell and strain was measured using a 50 mm extensometer (Figure 7b).

Figure 7.

Full-size tensile testing: (a) test setup and (b) full-size coupon with extensometer attached.

High-speed actuator tensile coupon tests

The displacement rates required for this analysis far exceeded the capabilities of the MTS universal testing machine used for the static tests. Therefore, a new test setup was devised using a recently acquired high-speed servo-hydraulic actuator (MTS244.21). This actuator, with a piston area of 25.2 cm² and a specified peak valve flowrate of 5.67 L/s, was theoretically capable of reaching speeds of 2.25 m/s. Combined with the subsize coupon geometry described previously, a theoretical strain rate of 90 s⁻¹ could be reached with this actuator.

Test setup

Unlike the MTS245, the high-speed actuator was not part of a built-in test frame. The actuator first had to be attached to a special test frame purpose-built for the actuator and connected to the laboratory strong floor (Figure 8a). A challenge with this installation was the large space required for a slack adaptor (Figure 8b). The slack adaptor test setup was similar to that used by Yu and Jones (1991) and Luecke et al. (2005). A slack adaptor was required for the high-speed tests to allow the actuator to reach a specified speed prior to loading the tensile coupon. Doing so achieves an almost-uniform strain rate for the duration of the loading up to yield, during the test, under ideal conditions. Without a slack adaptor, due to the subsize coupon gauge length, the coupon would deform significantly and possibly fracture before the desired strain rate was achieved.

Figure 8.

High-speed tensile testing: (a) test setup, (b) slack adaptor schematic, and (c) coupon schematic.

The test setup devised did not feature hydraulic clamps like the MTS245 universal testing machine used to conduct the static tests. Therefore, in order to generate the necessary grip force, small 50 × 50 mm plates were welded to the ends of the subsize coupons (Figure 8c). These blocks allowed the necessary force transfer from the actuator to the coupon through bearing. For corner coupons, the curved corner shape was maintained and the centroid of the coupons was aligned with the centroid of the blocks during welding.

Instrumentation

As the speed of loading increases, it becomes more difficult to reliably record accurate data. Therefore, when devising the instrumentation plan, care was taken to ensure that instrumentation would work for all displacement rates, from the “static” tests to the highest speed. The final instrumentation setup included load, displacement, and up to four channels of strain. Data were recorded using two modules (four channels each) of a Quantum X MX410 data acquisition system. This setup is capable of recording data at rates up to 96000 Hz. In order to capture the high-speed load-displacement data, all tests were recorded at this maximum rate with the exception of the “static” tests. The static reference tests were recorded at 5 Hz.

All coupons had at least two strain gauges, one on each face, installed on the gauge length of the coupon. These strain gauges were necessary to record an accurate strain time history since the extensometer used for the full-size static tests was not practical for the intermediate strain rate tests. Post-yield strain gauges manufactured by Tokyo Sokki Kenkyujo (TML), the YFLA series, were used for measuring strain in the gauge length of the coupons. The gauges (model YFLA-5-3LT) have a length and width of 5 and 1.9 mm, respectively. Two strain gauges on opposite faces were necessary for calculation of an average axial strain, thus enabling the removal of any bending strain from the strain results. Eccentricities in the load path, caused by curvature in the RHS and minor test setup imperfections, are the common cause of bending.

Due to the need to record the very short-duration loading, the built-in MTS load cell was replaced by an Interface miniature 45 kN load cell. This load cell was chosen because of its high natural frequency (22 kHz) that aids in the capture of the short-duration loading without excitation of its natural frequency. Load pulses with frequencies approaching the natural frequency of the load cell cause undesirable oscillation of the signal. However, despite the selection of this load cell, some of the rapid displacement rates used for this test program caused the load cell to exhibit excitation of its natural frequency, confounding the recorded load-time history. Therefore, at the higher displacement rates, two strain gauges, one on each face, were added to the bottom grip of the coupon to accurately measure the load-time history. These grips tended to remain elastic during a tensile coupon test due to their larger cross-sectional area, therefore allowing grip strain to be linearly related to grip stress. A similar procedure was used by Luecke et al. (2005) and Yu and Jones (1991) to determine load at high displacement rates. Standard TML FLA series strain gauges were used for the grip locations. Figure 9 illustrates the strain gauge locations on the tensile coupon.

Figure 9.

Strain gauge location and nomenclature.

Data processing

All data analyses used the standard 0.2% offset method to determine yield stress and yield strain. Measured strain rates over the strain range from zero to yield were consistently nonlinear and required a linear curve fit to determine the average strain rate over this range. For each average YFLA strain time history, a linear curve fit was conducted from zero to the yield strain and the slope of this line was reported as the strain rate, as is indicated in UFC 3-340-02 (DOD, 2008). Figure 10 illustrates a typical set of strain versus time histories and the corresponding linear strain rate curve fit. This method was chosen to align the results with the current practice in protective design.

Figure 10.

Typical strain versus time with linear strain rate fit for high-speed tensile coupons.

As the rate of testing increased and the load pulse frequency approached the natural frequency of the load cell, the quality of the load cell data decreased. At displacement rates of approximately 1 m/s, the error in the load cell signal began to significantly obscure the true load-time history. Thus, beyond 1 m/s, the FLA strain gauges installed on the grips were used to obtain force data. To calculate load from the grip strain data, the elastic modulus and grip area are needed. The cross-sectional grip area was easily measured using calipers for the flat coupons and was determined by weighing a fixed length of coupon grip and assuming a density of steel of 7850 kg/m³ for the corners.

The elastic modulus was calculated using the load cell data when possible. For the lower displacement rates, the elastic modulus was determined by fitting a linear curve to the stress-strain data over the linear elastic range (typically 0 to 0.001 strain). When the load cell data lost their accuracy, an alternative method was used, which involved the load cell data and the grip strain data. The load cell force was divided by the constant cross-sectional grip area to convert it to grip stress data. This stress data was then divided by the accompanying strain value at each time step. For the majority of the coupons, it was possible to fit a constant line through the data to determine the elastic modulus (see Figure 11). However, the elastic modulus does not change with strain rate (DOD, 2008), so it was also acceptable to use the average value over different strain rates. The grip strain, calculated grip area, and the elastic modulus were then used to determine the force.

Figure 11.

Calculation of the elastic modulus for a typical intermediate strain rate test (120-06F shown).

Results and discussion

Static tensile coupon comparison

The accuracy of the high-speed actuator test setup was verified by comparing the static subsize coupon test results with those gathered from the full-size coupon tests completed on the MTS245 universal testing machine. The slack adaptor was not needed for the subsize static tests, so the actuator was positioned such that the adaptor provided no slack. Static tests were run at a displacement rate of 0.003 mm/s, which corresponds to the 0.006 mm/s used for the full-size coupons with double the gauge length to achieve a consistent strain rate. Figures 12 and 13 illustrate a comparison of the results for the full-size and sub-size coupons for the flats and corners respectively.

Figure 12.

Typical static stress versus strain curves for flat coupons (RHS 120 × 120 shown).

Figure 13.

Typical static stress-strain curves for corner coupons (RHS 120 × 120 shown).

Table 5 summarizes and compares the average yield stress values, obtained using the 0.2% offset method, and average ultimate stress values, for each coupon type.

Table 5.

Measured static full-size and sub-size coupon material properties.

Specimen	Location	Yield stress (MPa)			Ultimate stress (MPa)
Specimen	Location	Full-size	Subsize	% diff.	Full-size	Subsize	% diff.
RHS 120 × 120 × 6	Flat	423	425	+0.5	510	510	0.0
RHS 120 × 120 × 6	Corner	530	533	+0.6	568	586	+3.2
RHS 80 × 80 × 3	Flat	435	435	0.0	527	536	+1.7
RHS 80 × 80 × 3	Corner	528	549	+4.0	575	605	+5.2
RHS 60 × 60 × 3	Flat	446	452	+1.3	534	557	+2.0
RHS 60 × 60 × 3	Corner	507	547	+7.9	553	613	+10.8

MPa: megapascals; RHS: rectangular hollow section.

As evidenced by Figures 12 and 13, and Table 5, the full-size and sub-size setups show good agreement when comparing the yield and ultimate stresses. Based on these findings, the validity of the sub-size test setup for the high-speed tensile tests was confirmed.

Dynamic tensile coupon results

Once the test setup was established for the dynamic testing, 65 sub-size coupons were prepared for high-speed testing. These coupons were divided into six “bands,” with each band aiming for a particular strain rate. As stated previously, each band contained four coupons (two from flats, two from corners) from each of the three RHS sizes. The bands were tested at escalating displacement rates, achieving strain rates from 0.1 to 17.9 s⁻¹. Figure 14 illustrates a typical set of stress versus strain curves from the six dynamic bands as well as the static curve for flat coupons, from the RHS 120 × 120 × 6 specimen. The stress-strain curves are truncated at the “failure” of the YFLA strain gauge data. This failure was typically due to delamination of one or both strain gauges from the coupon face. However, failure typically occurred at a strain much higher than the yield strain and hence did not affect the results.

Figure 14.

Stress-strain results at various strain rates for select RHS 120 × 120 flat coupons.

Although there is scatter in the results, Figure 14 illustrates an increase in both yield stress and ultimate stress due to elevated strain rate. The dynamic yield and ultimate stresses were compared against the sub-size static values to determine the DIFs for yield stress (DIF_y) and ultimate stress (DIF_u). Comparisons were conducted based on the location of the coupon within the RHS specimen. For example, static values from coupon 120-01B were used to calculate the DIFs for dynamically tested coupons from location 120-B (e.g., 120-06B). Tables 6 and 7 detail the stresses and DIFs for the tested coupons.

Table 6.

Key flat coupon test results.

Specimen	Strain rate (s⁻¹)	f_y (MPa)	f_dy (MPa)	DIF_y	f_u (MPa)	f_du (MPa)	DIF_u
120-01B	1.135E-05	433	N/A	1.00	515	N/A	1.00
120-01F	6.446E-06	418	N/A	1.00	504	N/A	1.00
120-02B	0.1144	433	506	1.17	515	580	1.13
120-02F	0.1024	418	458	1.10	504	567	1.12
120-03B	1.042	433	534	1.24	515	611	1.19
120-03F	1.588	418	475	1.14	504	602	1.19
120-04B	3.75	433	592	1.37	515	622	1.21
120-04F	4.26	418	496	1.19	504	604	1.20
120-05B	9.27	433	568	1.31	515	645	1.25
120-05F	11.12	418	491	1.17	504	630	1.25
120-06B	10.68	433	573	1.33	515	667	1.29
120-06F	10.77	418	524	1.25	504	658	1.30
120-07B	14.55	433	522	1.21	515	651	1.26
120-07F	13.46	418	500	1.19	504	641	1.27
80-01B	1.345E-05	416	N/A	1.00	522	N/A	1.00
80-01F	9.526E-06	453	N/A	1.00	549	N/A	1.00
80-02B	0.0784	416	476	1.14	522	594	1.14
80-02F	0.0874	453	517	1.14	549	637	1.16
80-03B	1.268	416	505	1.21	522	628	1.20
80-03F	1.182	453	512	1.13	549	653	1.19
80-04B	3.12	416	566	1.36	522	615	1.18
80-04F	3.93	453	581	1.28	549	635	1.16
80-06B	7.32	540	502	1.21	594	653	1.25
80-06F	15.18	558	519	1.14	616	705	1.28
80-07B	15.62	540	471	1.13	594	688	1.32
80-07F	13.51	558	483	1.07	616	728	1.32
60-01B	1.198E-05	436	N/A	1.00	551	N/A	1.00
60-01F	1.312E-05	467	N/A	1.00	563	N/A	1.00
60-02B	0.1048	436	482	1.11	551	597	1.08
60-02F	0.0859	467	504	1.08	563	606	1.08
60-03B	1.321	436	507	1.16	551	626	1.14
60-03F	0.961	467	537	1.15	563	637	1.13
60-04B	4.78	436	507	1.16	551	622	1.13
60-04F	3.40	467	602	1.29	563	635	1.13
60-05B	8.67	436	574	1.32	551	670	1.22
60-06B	17.41	436	580	1.33	551	686	1.25
60-06F	11.44	467	547	1.17	563	702	1.25
60-07B	7.23	436	771	1.77	551	701	1.27
60-07F	16.50	467	527	1.13	563	734	1.30

MPa: megapascals; DIF: dynamic increase factor.

Table 7.

Key corner coupon test results.

Specimen	Strain rate (s⁻¹)	f_y (MPa)	f_dy (MPa)	DIF_y	f_u (MPa)	f_du (MPa)	DIF_u
120-01C	9.415E-06	537	N/A	1.00	592	N/A	1.00
120-01G	1.226E-05	528	N/A	1.00	579	N/A	1.00
120-02C	0.1267	537	577	1.07	592	647	1.09
120-02G	0.1141	528	578	1.09	579	644	1.11
120-03C	1.015	537	606	1.13	592	684	1.15
120-03G	1.185	528	591	1.12	579	675	1.16
120-04C	3.22	537	658	1.23	592	692	1.17
120-04G	4.16	528	596	1.13	579	682	1.18
120-05C	8.38	537	628	1.17	592	738	1.25
120-05G	9.37	528	657	1.24	579	736	1.27
120-06C	12.58	537	614	1.14	592	746	1.26
120-06G	16.35	528	623	1.18	579	720	1.24
120-07C	13.52	537	645	1.20	592	732	1.24
120-07G	17.89	528	598	1.13	579	719	1.24
80-01A	1.280E-05	540	N/A	1.00	594	N/A	1.00
80-01E	1.074E-05	558	N/A	1.00	616	N/A	1.00
80-02A	0.1093	540	588	1.09	594	657	1.10
80-02E	0.0706	558	600	1.08	616	678	1.10
80-03A	0.806	540	631	1.17	594	681	1.15
80-03E	1.029	558	585	1.05	616	695	1.13
80-04A	3.28	416	669	1.24	522	716	1.21
80-04E	3.16	453	604	1.08	549	703	1.14
80-06A	10.50	416	666	1.23	522	750	1.26
80-06E	14.35	453	773	1.39	549	781	1.27
80-07A	15.06	416	570	1.06	522	782	1.32
80-07E	10.44	453	726	1.30	549	795	1.29
60-01A	1.096E-05	555	N/A	1.00	623	N/A	1.00
60-01C	1.349E-05	539	N/A	1.00	603	N/A	1.00
60-02A	0.1054	555	617	1.11	623	693	1.11
60-02C	0.0919	539	583	1.08	603	657	1.09
60-03A	1.096	555	598	1.08	623	696	1.12
60-03C	1.042	539	586	1.09	603	684	1.13
60-04A	3.36	555	634	1.14	623	711	1.14
60-04C	3.91	539	601	1.11	603	682	1.13
60-06A	11.81	555	762	1.37	623	753	1.21
60-06C	10.39	436	696	1.29	551	756	1.25
60-07A	12.12	555	798	1.44	623	797	1.28
60-07C	15.55	436	706	1.31	551	763	1.26

MPa: megapascals; DIF: dynamic increase factor.

Yield DIF

The majority of existing material research for blast focuses on the yield DIF_y as the dynamic yield stress is the primary material property that has the prime influence over the displacement response of an element.

Figures 15 to 17 illustrate the DIF_y relationship with strain rate for the three different RHS sizes. Each figure includes a Cowper–Symonds curve fit for the flat and corner coupons separately. The flats and corners are treated differently due to their residual stress profiles, as indicated in Sun and Packer (2014). Flats exhibit stronger dynamic increases than corners due to a reduced residual stress profile compared with the corners. All curve fits were completed using the Matlab Curve Fitting Tool (Mathworks, 2016) and a second order power function. The Trust-Region algorithm was used to fit the Cowper–Symonds curves and the R² values recorded.

Figure 15.

DIF_y versus strain rate with Cowper–Symonds curve fit for RHS120 × 120 coupons.

Figure 16.

DIF_y versus strain rate with Cowper–Symonds curve fit for RHS80 × 80 coupons.

Figure 17.

DIF_y versus strain rate with Cowper–Symonds curve fit for RHS 60 × 60 coupons.

Due to the higher residual stresses, and a higher initial yield stress, a lower DIF_y is expected in the corners, which was the general trend observed. Since the goal of the research is a generalized set of parameters for RHS, all of the coupon results were then combined and plotted together to determine a combined set of Cowper–Symonds parameters (Figure 18).

Figure 18.

DIF_y versus strain rate with Cowper–Symonds curve fit for all tensile coupons.

The Cowper–Symonds parameters for the eight curves are summarized in Table 8 along with their R² fit values. The results indicate the variability that can be seen when determining these parameters. Not only is the difference pronounced between the flat and corner coupons, but also between the various RHS sizes. The large scatter can be attributed to material variability and also to the difficulty with conducting and analyzing high-speed tests. “Noise” in the recorded data can have a large influence at the higher strain rates.

Table 8.

Material model parameters.

Specimens	Cowper–Symonds			Johnson–Cook
Specimens	C (s⁻¹)	q	R ²	c	R ²
RHS 120 × 120 flat coupons	208,201	7.199	0.625	0.01763	0.651
RHS 80 × 80 flat coupons	11,046	5.711	0.594	0.01977	0.568
RHS 60 × 60 flat coupons	5113	5.160	0.339	0.01994	0.311
RHS 120 × 120 corner coupons	1,006,574	6.614	0.730	0.01222	0.742
RHS 80 × 80 corner coupons	6614	4.296	0.501	0.01414	0.423
RHS60 × 60 corner coupons	94	1.877	0.852	0.01726	0.507
All flat coupons	19,609	5.938	0.447	0.01904	0.434
All corner coupons	4168	4.018	0.575	0.01435	0.476
All flat specimens, including Sun and Packer (2014)	26,583	5.995	0.724	0.02405	0.579
All corner specimens, including Sun and Packer (2014)	24,405,134	8.554	0.364	0.01457	0.354

RHS: rectangular hollow section.

The Johnson–Cook strain rate function was also fitted to the data and the strain rate parameter, c is listed in Table 8. A user-defined reference strain rate ( ${\overset{\cdot}{ε}}_{0}$ ) of 1.0 × 10⁻⁵ s⁻¹, the strain rate of the static tests, was used for the Johnson–Cook fits. Like the Cowper–Symonds fits, the Trust-Region algorithm was used in the Matlab Curve Fitting Tool and the R² values were recorded. With the exception of the RHS 120 × 120 coupons, the Johnson–Cook model is a worse fit than the Cowper–Symonds model to the recorded data. A typical example of the Johnson–Cook curve fit is shown in Figure 19.

Figure 19.

DIF_y versus strain rate with Johnson–Cook curve fit for all tensile coupons.

Comparison with other RHS data

In order to better cover the blast strain rate spectrum and to further generalize the material model for RHS, the results of the tensile coupon tests described herein are compared with the results of Sun and Packer’s (2014) tensile SHPB tests. This then creates a dynamic cold-formed RHS data set that spans intermediate and high strain rates from 0.1 to 1000 s⁻¹. Figure 20 incorporates the work of Sun and Packer, along with a set of Cowper–Symonds curves for the combined data set. Figure 21 illustrates the same data with a set of Johnson–Cook curves for the combined data.

Figure 20.

DIF_y versus strain rate with Cowper–Symonds curve fit for all cold-formed RHS specimens, including Sun and Packer (2014) data.

Figure 21.

DIF_y versus strain rate with Johnson–Cook curve fit for all cold-formed RHS specimens, including Sun and Packer (2014) data.

Table 8 also includes the Cowper–Symonds and Johnson–Cook parameters for the combined data set. For almost all the tensile Cowper–Symonds curve fits, there is an increase in both C and q from the corners to the flats (the exception being C for RHS 120 × 120). The addition of the Sun and Packer (2014) data flattened out the Cowper–Symonds curve fit for the corner coupons significantly. This causes both C and q to increase from the flats to the corners for the combined data set. The combined flat specimen curve fit (“all flat specimens, including Sun and Packer”) is very similar to the curve fit for the “all flat coupons” alone, which indicates a reasonable agreement between the two data sets. For the Johnson–Cook curve fits, there is only a very small change in the slope for the corners, whereas there is a noticeable increase in the slope for the flats. As evidenced by the R² values, the Johnson–Cook material model is a worse fit for the combined data set.

The high level of variance in the material model parameters determined for cold-formed RHS tests underscores the need for this test program. The parameters in Table 8 differ significantly from those previously determined for other steels in Tables 2 and 3. This difference confirms the importance of testing across the whole spectrum of strain rates and does not rely on only the extreme 100–1000 s⁻¹ range to calibrate models. The material model parameters from the combined data set, covering a strain rate range of 0.1–1000 s⁻¹, are best suited for future use in numerical modeling of RHS subject to elevated strain rate loading.

Comparison with previous investigations

Figure 22 plots the combined RHS Cowper–Symonds curves against previously plotted curves from Figure 2.

Figure 22.

DIF_y versus strain rate for various Cowper–Symonds model parameters, with combined RHS results.

The RHS curves are quite different to the majority of the curves, with the exception of those determined by Yu and Jones (1991), who tested mild steel with a yield stress of approximately 250 MPa. Previous investigations (Ritchie et al., 2015) have identified the influence that these parameters can have on FE modeling results for RHS specimens subject to blast loading using LS-DYNA (LSTC, 2014). The Cowper–Symonds parameters determined herein are best suited for use in implicit FE modeling of RHS under blast loading in the future. The same can be said for the Johnson–Cook parameters, as illustrated in Figure 23.

Figure 23.

DIF_y versus strain rate for various Johnson–Cook model parameters, with combined RHS results.

Ultimate DIF

Generally, the use of the Cowper–Symonds curve has been limited to DIF_y. However, for comparative purposes, Figure 24 illustrates the DIF_u dependence on strain rate as well as the Cowper–Symonds curves for the flat and corner tensile coupons.

Figure 24.

DIF_u versus strain rate with Cowper–Symonds curve fit for all tensile coupons.

There is noticeably less scatter in the DIF_u data when compared with the DIF_y data (Figure 18). This can likely be attributed to the differing methodologies used for determining the dynamic yield and ultimate stresses. Yield stresses often relied on assumptions for the elastic modulus and the dynamic yield stress calculated using the 0.2% offset method, which is more susceptible to data “noise.” The dynamic ultimate stress calculation is more straightforward and only relied on filtering of the load cell data at higher strain rates. This leads to improved accuracy. Table 9 lists the Cowper–Symonds parameters for the ultimate stress curve fits, along with the R² values. The latter are high, confirming the low scatter in the DIF_u results. Table 9 also includes the Johnson–Cook strain rate parameter and R² values.

Table 9.

Material model parameters for DIF_u.

Specimens	Cowper–Symonds			Johnson–Cook
Specimens	C(s⁻¹)	q	R ²	c	R ²
All flat coupons	7793	4.861	0.889	0.01701	0.805
All corner coupons	4593	4.331	0.926	0.01599	0.798

DIF_u: dynamic increase factor.

The other noticeable difference is that the flat and corner curves are much closer together than for the yield stress, as illustrated in Figure 25. This is expected because the residual stresses caused by cold-forming do not significantly alter the ultimate stress values. Since the static reference point for the flat and corner dynamic ultimate stress is similar, the DIF_u results are much closer.

Figure 25.

Comparison of DIF_y and DIF_u Cowper–Symonds curve fits for all tensile coupons.

Conclusion

The elevated strain rate behavior of cold-formed steel RHSs was investigated by initially performing large-scale field air-blast experiments on RHS members. Intermediate strain rate properties of the RHS material were then determined in the laboratory by using a high-speed actuator to test tensile coupons taken from the flat and corner regions of different-size members. The dynamic material properties were investigated to better calibrate elevated strain rate steel material models for use in protective design and analysis. The intermediate strain rate results were combined with previous RHS tests at high strain rates to produce a unified model for DIF_y and DIF_u applicable to RHS steel, subject to a wide range of impulsive strain rates. The behavioral model differs from others previously produced for general structural steel and is best suited for use with cold-formed RHS specimens in the future.

Footnotes

Appendix

Acknowledgements

Technical advice and assistance from Prof. M.V. Seica, Prof. D.Z. Yankelevsky, and Mr. F. Wei are also highly appreciated.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors are appreciative of the financial aid and in-kind support of the Explora Foundation to the University of Toronto “Centre for Resilience of Critical Infrastructure.” Financial support has also been received from the Natural Sciences and Engineering Research Council of Canada (NSERC), the Steel Structures Education Foundation (SSEF), the Thornton Tomasetti Foundation, the Lyon Sachs Graduate Research Fund, the Australian Research Council (ARC) Discovery Grant (Project ID: DP130100181), and the Tsinghua Initiative Scientific Research Program (No. 20131089347).

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