冲击作用下框架结构的连续性倒塌性能

宿华祥; 易伟建; 黄义谋

doi:10.11858/gywlxb.20190806

摘要: 借助ANSYS/LS-DYNA软件建立了钢筋混凝土框架的有限元模型，研究了钢筋混凝土框架在冲击荷载作用下的连续性倒塌性能，冲击体质量为1 000 kg，冲击速度为4 m/s。通过对钢筋混凝土构件冲击试验和框架倒塌过程的验证，保证了数值模拟的有效性。分析结果表明：冲击中柱后结构倒塌过程中，有“拱作用”向“悬索作用”转换的机制，中柱顶部位移先向上后向下，边柱顶部位移先向外后向内；同样冲击作用下，柱轴力越小，则抗冲击能力越强，不同的偏压作用对柱的抗冲击性能的影响不同；加密柱箍筋能够增强钢筋混凝土柱的抗冲击能力，延缓甚至避免钢筋混凝土框架结构的连续性倒塌。

关键词:

Abstract: In order to study the continuous collapse behavior of reinforced concrete frame under impact load, the finite element model of reinforced concrete frame is established by ANSYS/LS-DYNA. The impact mass is 1 000 kg and the impact velocity is 4 m/s. The effectiveness of the numerical simulation is guaranteed by the verification of the of reinforced concrete members impact experiment and the frame collapse process. The analysis draws following conclusions: in the process of structure collapse of the mid column impacted, there is a mechanism of “arch effect” to “suspension effect”, the top of the middle column first goes up and then down, and the top of the side column first goes out and then inward; under the same impact load, the smaller the column axial force, the stronger column impact resistance and the column impact resistance is affected by different eccentric pressure. The impact resistance of reinforced concrete columns can be enhanced by infill column stirrups which can delay or even avoid the reinforced concrete frame structures continuous collapse.

Key words:

The corrugated interface between different fluids grow when accelerated from a low-density fluid to a high-density fluid, which is called Rayleigh-Taylor (RT) instability^[1-2].This phenomenon may also occur in metals, but usually under a high pressure and at a high strain-rate, but differs most distinctly from the RT instability in fluids in its strength effect of the metal, which can stabilize the perturbation growth^[3-4] and make the metallic RT instability more complex and difficult.Here, it is also affected by the loading state and the properties of the metallic materials.The metallic RT instability at high pressure and strain-rate can be observed in inertial confinement fusion^[5], supernova explosion^[6], asteroid collision^[7], the motion of earth's inner core and plate tectonics^[8], and so on.Therefore, the metallic RT instability is currently a major concern for researchers and receives a great deal of academic attention.

In theoretical studies about the metallic interface instability, dispersion relations of the perturbation growth are derived mainly based on the energy^[9-11] or force equilibrium^[12-13].However, the previous linear analysis cannot predict the perturbation growth accurately just by applying the perfect plastic constitutive relation and constant pressure loading state.Based on the energy balance, a perturbation growth equation using Steinberg-Guinan (SG) and the Johnson-Cook constitutive models, as well as a variable pressure loading process consistent with experiments, has been derived that precisely predicts the growth of metallic RT instabilities driven by detonation and laser plasma.However, the linear analysis still has its limitations and does not take full account of the loading procedure.

Experimental studies of the metallic RT instability started in the 1970s.The pioneering experimental research^[14] was the perturbation growth of a flat aluminum plate accelerated by the expansion of detonation products, which was observed using a high-energy X-ray facility.What was achieved then inspired researchers, and the similar equipment was utilized in later research^[15-17].In the USA and Russia particularly, numerous numerical simulations and experimental investigations for the metallic RT instability have been carried out, but have mainly concentrated on the perturbation growth and such influencing factors as the initial amplitude, the wavelength and material properties.Igonin and Ignatova et al.^[18-19] experimentally and numerically studied the dynamic behaviors of copper (Cu) and tantalum (Ta) subjected to both shock and shockless loading by employing a perturbation growth method.They observed that the formation of the bi-periodic twin structures resulted in an initial loss of the shear strength of Cu, but failed to observe localization in Ta.Olson et al.^[16] experimentally studied the effects of the grain size and material processing on the RT perturbation growth of Cu.They found that both the single-crystal orientation and the strain hardening due to the material processing can affect the perturbation growth, but the polycrystalline grain size cannot.For the plane detonation, the loading pressure is generally about 30 GPa.To enhance the loading pressure, Henry de Frahan et al.^[17] studied the beryllium RT instability using an iron flyer plate to impact the second high explosive (HE) to raise the pressure to 50 GPa in their experiments, and combined numerical simulations to calibrate the feasibility of different constitutive models.When the sample is driven by electromagnetism^[20-21] or laser^{[4, 22]}, the loading pressure can be further increased.Very extreme conditions of pressures over 1 000 GPa and strain rates of 10⁸ s^-1 have been achieved at the National Ignition Facility, USA, where the RT instability experiment in vanadium was carried out, and constitutive models in solid phase were tested by comparing simulations with experiments measuring the perturbation growth^[23] under the extreme conditions mentioned.

In the metallic interface instability, the perturbation growth is related to and arrested by the material strength.Moreover, some investigations have demonstrated that the material strength increases under these extreme conditions.Results from the metallic RT experiments and computations by Barnes et al.^[14] show that the yield strength of 1100-0 aluminum is over 10 times larger than the standard parameter, and the yield strength of 304 stainless steel also increases by more than three times.Using the SG constitutive strength model, calculations of plasma-driven quasi-isentropic RT experiments of Al-6061-T6 using the Omega laser at a peak drive pressure of 20 GPa indicate that its yield strength is a factor of about 3.6 times over the ambient value^[22].In Park et al.'s^[4] plasma-driven quasi-isentropic polycrystalline vanadium RT experiments using the Omega laser with a peak drive pressure of 100 GPa, the measured RT growth was substantially lower than predictions using the existing constitutive models (SG and Preston-Tonks-Wallace) that work well at low pressures and long timescales.Using the SG model, the simulations agree with the RT experimental data when the initial strength is raised by a factor of 2.3.Therefore, the SG and Preston-Tonks-Wallace models underestimate the strength of vanadium under very high pressures and strain rates.Belof et al.^[24] first measured the dynamic strength of iron undergoing solid-solid phase transition by using RT instability.In conjunction with detailed hydrodynamic simulations, the analysis results revealed significant strength enhancement of the dynamically generated ε-Fe and reverted α′-Fe, comparable in magnitude to the strength of austenitic stainless steels.Therefore, the metallic RT instability was suggested and used as a tool for evaluating the material strength of solids at high pressures and high strain rates^{[3, 25]}, and then for modifying or developing new constitutive models for these conditions^[26-27].

In view of the dominant role of the material strength in metallic interface instabilities, and the limitations of existing constitutive models at high pressures and high strain rates, we aimed to investigate the material strength and its effects on metallic interface instabilities.In this paper, we also conducted an RT instability experiment in explosion-driven aluminum, and measured the perturbation growth using X-ray radiography.In combination with elastic-plastic hydrodynamic simulations, we investigated the dynamic behavior of metallic RT instabilities and the role of the material strength in these.

1. Experimental Setup

Following that of Barnes et al., ^[14]our experiment used the setup as shown in Fig. 1(a), where we have a sketch of the experimental setup consisting of a detonator, a booster, plane wave lens, JO-9159 HE (100 mm in diameter and 50 mm in thickness), an aluminum sample, and a vacuum.Fig. 1(b) shows the experimental sample of aluminum with a diameter of 65 mm and a thickness of 1.5 mm in the central region.An initial sinusoidal perturbation was machined on the side of the aluminum sample facing the HE.The perturbation amplitude and wavelength were 0.25 mm and 6 mm, respectively.The HE products crossed the void of 3.5 mm between the sample and HE and accumulated on the perturbation interface of the sample, providing a smooth rise to peak pressure and a quasi-isentropic drive.Moreover, the void between the sample and HE can ensure that the temperature of the sample at high pressures remain below the melting point^[22].

Figure 1. Sketch of the experimental setup and sample

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In the experiment, X-ray radiography was used to record the evolution of the perturbation interface from the JO-9159 explosive detonation at zero time.A Doppler pin system was used to measure the history of the free surface velocity, which can be integrated to obtain the corresponding free surface displacement.Fig. 1(c) shows the distribution of measurement points of the free surface velocity, where both points 1 and 2 correspond to the wave trough positions with one wavelength interval, and point 3 corresponds to the wave crest position with 1.5 wavelength intervals from point 2.

2. Numerical Methods

Based on our indoor hydrodynamic code of compressible multi-viscous flow and turbulence (MVFT)^[28-30], we developed a detonation and shock dynamics code with high fidelity by considering the explosive detonation and the elastic-plastic behavior of the material.This code can be used to study the physical problem of multi-materials, large deformation, and strong shock.The governing equations in conserved form are as follows

$\left\{ \begin{array}{l} \frac{\partial }{{\partial t}}\int_V {\rho {\rm{d}}V} =-\oint_S {\rho {u_i}{n_i}{\rm{d}}S} \\ \frac{\partial }{{\partial t}}\int_V {\rho {u_j}{\rm{d}}V} =-\oint_S {P{n_j}{\rm{d}}S}-\oint_S {\rho {u_i}{u_j}{n_i}{\rm{d}}S} + \oint_S {{s_{ij}}{n_i}{\rm{d}}S} \\ \frac{\partial }{{\partial t}}\int_V {\rho E{\rm{d}}V} = - \oint_S {{u_i}P{n_i}{\rm{d}}S} - \oint_S {\rho {u_i}E{n_i}{\rm{d}}S} + \oint_S {{s_{ij}}{u_j}{n_i}{\rm{d}}S} \end{array} \right.$

(1)

where i and j represent the x, y, and z directions; V is the control volume, S the surface of control volume, n the unit vector of the external normal direction, ρ, u_k (where k=i, j), p, and E are the density, velocity, pressure, and total energy of per unit mass; and s_ij the deviation stress tensor.

The physical problem as described by Eq.(1) was decomposed into three one-dimensional problems.For each of them, the physical quantities in a cell were interpolated and reconstructed using a piecewise parabolic method (PPM).The one-dimensional problem was then resolved using a two-step Euler algorithm:First the physical quantities were solved by the Lagrange matching, and then remapped back to the stationary Euler meshes.The effect of material strength, explosive detonation, and artificial viscosity were implemented in the Lagrange step.The multi-material interface was captured by applying a volume-of-fluid (VOF) method.

2.1 Equation of State

In our numerical simulations, the equation of state (EOS) for the explosive and aluminum are the Jones-Wilkins-Lee (JWL) and Mie-Grüneisen equations of state, respectively.The Jones-Wilkins-Lee equation of state is

$p\left( {\rho, T} \right) = A\left( {1-\frac{\omega }{{{R_1}v}}} \right){{\rm{e}}^{-{R_1}v}} + B\left( {1-\frac{\omega }{{{R_2}v}}} \right){{\rm{e}}^{ - {R_2}v}} + \frac{{\omega \bar E}}{v}$

(2)

where v=ρ₀/ρ is specific volume; A, B, R₁, R₂, and ω are constants; and E is the internal energy per unit volume.Table 1 lists the JWL EOS parameters of the JO-9159 explosive.The Mie-Grüneisen equation of state is

Table 1. Equation of state parameters of JO-9159 explosive

ρ/(g·cm^-3)	p_CJ/GPa	D_CJ/(km·s^-1)	A/GPa	B/GPa	R₁	R₂	ω
1.86	36	8.862	934.8	12.7	4.6	1.1	0.37

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$p = \frac{{{\rho _0}{c^2}\mu \left[{1 + \left( {1-{\gamma _0}/2} \right)\mu-a{\mu ^2}/2} \right]}}{{{{\left[{1-\left( {{S_1}-1} \right)\mu-\frac{{{S_2}{\mu ^2}}}{{\mu + 1}} - \frac{{{S_3}{\mu ^3}}}{{{{\left( {\mu + 1} \right)}^2}}}} \right]}^2}}} + \left( {{\gamma _0} + a\mu } \right)\bar E$

(3)

where μ=ρ/ρ₀-1 is the relative compression, ρ₀ the initial density, c the sound velocity at zero pressure, γ₀ the Grüneisen coefficient, and a, S₁, S₂, and S₃ are constants (in Table 2).

Table 2. Mie-Grüneisen equation of state parameters of aluminum

ρ/(g·cm^-3)	c/(km·s^-1)	γ₀	a	S₁	S₂	S₃
2.703	5.22	1.97	0.47	1.37	0	0

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2.2 Constitutive Model

In our simulations, the elastic-plastic behavior of aluminum at high pressures and high strain rates was described using the SG constitutive model.The SG model introduces pressure, temperature, and strain-rate terms into the elastic-plastic constitutive equation, while the coupling effect of pressure and strain rate on flow stress was characterized by the separating variables.Additionally, as the flow stress in the SG model relies on pressure, there is a coupling relationship between the material constitutive equation and the equation of state, which indicates the feature of pressure hardening of metal under high pressure.The dynamic yield strength Y_SG and the shear modulus G determined by the SG model are expressed as

${Y_{{\rm{SG}}}} = {Y_0}{\left[{1 + \beta \left( {{\varepsilon _p} + {\varepsilon _i}} \right)} \right]^n}\left[{1 + Ap{\eta ^{-1/3}}-B\left( {T-300} \right)} \right]$

(4)

$G = {G_0}\left[{1 + Ap{\eta ^{-1/3}}-B\left( {T-300} \right)} \right]$

(5)

where Y₀ and G₀ are the initial yield strength and the shear modulus, respectively; β and n are the material strain hardening coefficient and the hardening index, respectively; A is the pressure hardening coefficient; η=ρ/ρ₀ is the material compression ratio; and B is the temperature softening coefficient (in Table 3).

Table 3. Steinberg-Guinan constitutive model parameters of aluminum

Y₀/GPa	Y_max/GPa	G₀/GPa	β	n	A/GPa^-1	B/(10^-3K^-1)
0.29	0.68	27.6	125	0.1	0.0652	0.616

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3. Results and Discussions

In our experiment, X-ray radiography recorded an image of the perturbed interface at 7.78 μs, as shown in Fig. 2(a), from which we obtained the amplitude of 0.77 mm simultaneously by image processing.In the simulations, the mesh size was 15.6 μm, and Tables 2 and 3 list the parameters of the Mie-Grüneisen EOS and the SG constitutive model of aluminum, respectively.

Figure 2. Comparisons of the perturbed interface between experiment and numerical simulations ((a) Experimental image, (b) Simulated image at normal strengths Y₀ and G₀, (c) Simulated image at 10 times the normal strengths Y₀ and G₀)

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Fig. 3 shows the pressure histories of the crest (solid line) and the trough (dashed line) on the loading surface, which increase continuously and smoothly in a short time and form an approximate quasi-isentropic drive.Afterwards, the expansion of the detonation products decelerates gradually, and the loading pressure on the interface rises slowly.However, the pressure at the trough ascends faster than that at the crest, and the peak pressure at the trough is also relatively larger, because the detonation products converges at the trough and diverges at the crest.The average peak pressure is about 25 GPa, and the strain rate reaches 10⁶ s^-1.The loading pressure then reduces gradually, which is attributed to the decrease of the expansion pressure of the detonation products and the unloading effect of the rarefaction wave reflecting from the free surface.

Figure 3. Pressure histories of crest and trough at the loading surface

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Fig. 4 shows several contours of local pressure (a), density (b), and temperature (c) at 6 different times after the arrival of the detonation products at the loading surface, from left to right and top to bottom, at 6.36, 6.5, 6.7, 6.9, 7.1 and 7.3 μs.They exhibit an evolution process of the perturbed interface and the interaction of the wave and the interface.The blue part of the temperature contour is the aluminum sample, and the sample temperature remains below 500 K and far below the melting point of 1 200 K, which indicates that the sample is in the elastic-plastic state all the time.

Figure 4. Contours of local pressure (a), density (b), and temperature (c) at 6.36, 6.5, 6.7, 6.9, 7.1, and 7.3 μs from left to right and top to bottom after the arrival of detonation products at the loading surface

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Fig. 2(b) shows an image of the sample at 7.78 μs when the initial yield strength Y₀ and the shear modulus G₀ are normal values.Fig. 5 shows a comparison of the perturbation amplitudes between the experiment and numerical simulations, where the square symbol corresponds to the experimental result, and the solid black line corresponds to the numerical results when Y₀ and G₀ are normal.The simulated amplitude is much larger than that in the experiment when using normal values of Y₀ and G₀.This is because the aluminum strengthens under such conditions, and the SG constitutive model underestimates its strength, which can suppress the perturbation growth.

Figure 5. Growth histories of the perturbation amplitude

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Fig. 6 shows the time histories of the free surface velocity (a) and displacement (b) at 3 measurement points (dot-dot-dashed lines:experiment; solid, dashed, and dotted lines:simulations), which agree well with each other.Therefore, the calculations of detonation of the explosive and the thermodynamic state of the sample are exact.

Figure 6. Time histories of the free surface velocity (a) and displacement (b)

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Fig. 7 shows the calculated time histories of the strain at the crest (solid line) and trough (dashed line) of the loading surface.The deformation at the trough is much larger than that at the crest because the trough of the sample is in a stronger tensile stress state, which is the main mechanism for the deformation of the perturbation interface.Fig. 8 shows the time histories of the dynamic yield strength at the crest (solid line) and trough (dashed line) of the loading surface, calculated using the SG constitutive model, similar to the profile of the loading pressure, which demonstrate that the material strength increased as did the loading pressure under a certain condition.

Figure 7. Time histories of strain at the crest and trough of the loading surface

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Figure 8. Time histories of yield strength at the crest and trough of the loading surface

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Moreover, when the loading pressure reaches a peak, the dynamic yield strength was up to 3 times that of the initial value.In fact, the normal SG model is generally calibrated by conventional Hopkinson and Taylor impacts experiment with a lower strain rate.Under the current loading conditions (loading pressure of 25 GPa and strain rate of 10⁶ s^-1), the strength is not great enough to suppress the perturbation growth.However, when the initial yield strength Y₀ and the shear modulus G₀ increase to 10 times that of the normal values, good agreement between the experiment and simulation is achieved, as shown in Fig. 2(c) for the perturbation interface and in Fig. 5 for the perturbation amplitude with the dashed line.Therefore, the material strength intensively stabilizes the perturbation growth.The dotted lines in Fig. 5 are fitted lines from the simulated results, indicating that the perturbation amplitude grows exponentially over time.

We studied the effect of the initial yield strength and the initial shear modulus of the material on the evolution and growth of the perturbed interface.Figs. 9(a), 10(a), and 11(a) show the calculated growth histories of perturbation amplitude, strain histories, and dynamic yield strength histories at the trough of the loading surface, respectively, when the initial yield strength is fixed at the normal value and as the initial shear modulus increases gradually.The growth of the perturbation amplitude exhibits no change even when the initial shear modulus increases to 10 times that of the normal value, which means that the initial shear modulus has no influence on the material deformation and does not affect the dynamic yield strength.

Figure 9. Growth histories of the perturbation amplitude for different values of initial shear modulus (a) and yield strength (b)

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Figure 10. Time histories of strain at the trough of the loading surface for different values of initial shear modulus (a) and yield strength (b)

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Figure 11. Time histories of yield strength at the trough of the loading surface for different values of initial shear modulus (a) and yield strength (b)

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Figs. 9(b), 10(b), and 11(b) show the numerical results when the initial shear modulus is fixed and the initial yield strength gradually increases to 10 times that of the normal value.These indicate that, with the increase of the initial yield strength, the dynamic yield strength also increases, the material deformation is retarded, and the perturbation growth is suppressed markedly.Therefore, the initial shear modulus of the material exerts no effect on the growth of the metallic RT instability within a certain range, while the initial yield strength has an obvious effect on it.

4. Conclusions

We have established an experimental setup and developed a numerical simulation method to investigate the RT instability in metallic materials driven by explosion.We also studied the RT instability in aluminum, and drew the following conclusions:

(1) The perturbation amplitude grows following an exponential law over time.

(2) When using the normal physical property parameters of aluminum, simulated evolution of the perturbed interface agrees with experiment only qualitatively, and there is a big quantitative difference between them because the aluminum strengthens under high pressures and at high strain rates, and the SG constitutive model underestimates its strength as being not great enough to suppress the perturbation growth.

(3) When the initial yield strength and the initial shear modulus increase to 10 times their normal values, the numerical and experimental results are in good agreement both qualitatively and quantitatively.The underlying physical mechanism is the stabilization effect of material strength on the perturbation growth.Moreover, the initial shear modulus has no influence on the perturbation growth within a certain range whereas the initial yield strength does influence it strongly.Therefore, the material strength dominates the evolution of the metallic RT instability.

图 1 试验框架尺寸、配筋与测试仪器布置（单位：mm）

Figure 1. Detailed drawing of test frame size, reinforcement and arrangement of test instruments(Unit: mm)

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图 2 钢筋混凝土框架有限元模型

Figure 2. Finite element model of reinforced concrete frame

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图 3 A-1、A-2梁跨中挠度时程曲线（a）和损伤比较（b）

Figure 3. Comparison of deflection time history curves (a) and damage (b) between A-1 and A-2 beams in midspan

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图 4 荷载曲线

Figure 4. Load curve

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图 5 框架模拟与试验对比

Figure 5. Frame simulation results vs. experiment

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图 6 框架倒塌过程中变形损伤

Figure 6. Damage deformation of the frame collapse

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图 7 框架模型及受荷

Figure 7. Frame model and loading

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图 8 中柱柱顶竖向位移时程曲线

Figure 8. Curve of vertical displacement vs. time at the top of central column

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图 9 A柱、E柱柱顶水平位移时程曲线

Figure 9. Curves of horizontal displacement vs. time at the top of the column A and E

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图 10 A柱、E柱水平位移平均值时程曲线

Figure 10. Curve of average horizontal displacement vs. time of the column A and E

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图 11 水平位移与中柱竖向位移关系曲线

Figure 11. Relationship between horizontal displacement and vertical displacement of central column

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图 12 框架在冲击作用下的倒塌过程

Figure 12. Collapse process of frame under impact

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图 13 柱子受到冲击作用后的框架响应

Figure 13. Frame response of columns under impact

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图 14 A柱冲击作用点水平位移时程曲线

Figure 14. Curve of horizontal displacement vs. time at impact point of the column A

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图 15 加密箍筋后框架最终变形

Figure 15. Final deformation of frame with encrypted stirrups

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图 16 加密箍筋后冲击点处X方向位移时程曲线

Figure 16. Curve of X-direction displacement vs. time at impact point after encrypted stirrups

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表 1 材料本构模型及参数^[8]

Table 1. Constitutive model and material parameters^[8]

Parts	Material model	Material parameters
Hammer	*MAT_ELASTIC	ρ = 7 800 kg/m³, E = 200 GPa, ν = 0.27
Concrete	*MAT_CSCM	ρ = 2 400 kg/m³, f_c = 25 MPa, d = 20 mm
Distributed reinforcement	*MAT_PLASTIC_KINEMATIC	ρ = 7 800 kg/m³, E = 200 GPa, ν = 0.27, f_y = 416 MPa, f_u = 526 MPa
Stirrups	*MAT_PLASTIC_KINEMATIC	ρ = 7 800 kg/m³, E = 210 GPa, ν = 0.27, f_y = 370 MPa

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表 2 梁跨中最大位移比较

Table 2. Largest displacement comparison

Beam No.	Maximum displacement/mm		Relative error/%
Beam No.	Experiment	Simulation	Relative error/%
A-1	81.0	82.8	2.22
A-2	74.0	74.0	0
A-3	83.6	90.6	8.37
A-4	89.5	86.6	−3.24

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表 3 不同失效时间对应的动力系数

Table 3. Dynamic coefficients at different failure time

t₁/s	μ	t₁/s	μ	t₁/s	μ	t₁/s	μ
0.001	2.000	0.020	1.832	0.060	1.014	0.150	1.128
0.005	1.989	0.040	1.426	0.061	1.003	∞	1.000
0.010	1.956	0.045	1.314	0.090	1.215

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表 4 框架倒塌范围汇总

Table 4. Summary of frame collapse scope

Column No.	Collapse scope
A	No collapse
B	AB span and BC span
C	All
D	CD span and DE span
E	DE span

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冲击作用下框架结构的连续性倒塌性能

doi: 10.11858/gywlxb.20190806

作者简介:
宿华祥（1992–），男，硕士研究生，主要从事钢筋混凝土结构抗冲击性能研究. E-mail：15575888135@163.com

通讯作者:
易伟建（1954–），男，博士，教授，主要从事混凝土结构基本理论研究. E-mail：wjyi@hnu.edu.cn

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