Simplified theoretical analysis and numerical study on the dynamic behavior of FCP under blast loads

Abstract

Precast concrete structures have developed rapidly in the last decades due to the advantages of better quality, non-pollution and fast construction with respect to conventional cast-in-place structures. In the present study, a theoretical model and nonlinear 3D model are developed and established to assess the dynamic behavior of precast concrete slabs under blast load. At first, the 3D model is validated by an experiment performed by other researchers. The verified model is adopted to investigate the blast performance of fabricated concrete panels (FCPs) in terms of parameters of the explosive charge, panel thickness, and reinforcement ratio. Finally, a simplified theoretical model of the FCP under blast load is developed to predict the maximum deflection. It is indicated that the theoretical model can precisely predict the maximum displacement of FCP under blast loads. The results show that the failure modes of the panels varied from bending failure to shear failure with the mass of TNT increasing. The thickness of the panel, reinforcement ratio, and explosive charges have significant effects on the anti-blast capacity of the FCPs.

Keywords

precast structure / fabricated concrete panel / blast resistance / theory model / empirical equation

Introduction

Over the last few decades, the precast structure has been applied in many countries with great development in construction industrialization. Precast concrete construction as a type of building industrialization has been significantly used in building and bridge construction due to the low environmental pollution and high construction speed [¹].

Apart from the above benefits, the connections of fabricated concrete components are always vulnerable and the concerned problem for the precast concrete structures (PCSs). If PCSs or fabricated components are subjected to blast load or explosion accidents, they may take localized blast damage which can cause a severe global collapse. Although PCSs have been built in some countries, limited investigations focused on the dynamic behavior and anti-blast capacity under impact load or explosion accident. Due to the rapid development of PCSs in the world as well as with the increase of buildings in cities, fabricated concrete panels (FCPs) and fabricated concrete columns (FCCs) of PCSs are vulnerable to blast loads. The dynamic behaviors of FCPs or FCCs under blast load should be studied for their applications in civil engineering conveniently.

In the past, many papers focused mainly on seismic capacities of PCSs and components under dynamic loading. Majorities studies investigated the mechanical performance and seismic resistance of PCSs under seismic loads to apply this type of building in the high-intensity area. Moreover, several researchers mainly paid attention to the blast performance of conventional reinforced concrete (RC) and composite structures (CS) [²–¹¹]. Several meshfree and enriched meshfree approaches have been proposed and used to carry out the high-velocity concrete fragmentation analysis, such as large deformation lagrangian meshfree method, particle hydrodynamics method and so on [¹²–¹⁶]. However, no research on the blast resistance capacity of PCSs, FCPs, and FCCs under explosion can be found in the open literature. Wu et al. [¹⁷] proposed a type of precast slender composite shear wall and experimentally studied, it was found that this shear wall can reduce energy dissipation capacity and increase lateral strength and deformation capacity. Xu et al. [¹⁸] investigated the seismic behavior of precast reinforced concrete shear walls (PRCSW) with grout-filled sleeves connection by QSTs with accurate boundary condition simulation of axial-flexure-shear loads. Yan et al. [¹⁹] investigated the dynamical behavior of precast concrete beams subjected to impact loads, it is indicated that the dynamic performance of the RC beam affected by the precast region might affect under drop weight loads.

As the above mentioned, only limited studies focused on the seismic response of fabricated concrete structures under drop weight load or vehicle collision. In the present study, a theoretical model (TM) and three-dimensional finite element model (FEM) of FCP are developed and established and applied to investigate blast behaviors of FCP under blast load. First, the FEM of FCP is validated by a previous experiment [²⁰]. Then, parametric analyses of FCP thickness, TNT charges, the strength of concrete and reinforcement bar (RB) on the blast resistance of panels are performed. Based on the numerical results, an empirical formula is proposed to predict maximum central deflection and blast resistance of FCPs. Finally, a TM is developed to predict the maximum deflection of FCP under blast load, which has good accuracy in evaluating the peak displacement under the blast load.

Constitutive model

Concrete

Mat-72-Rel3 is the release III of the K&C Concrete Model, which has been validated and widely used in the simulation of conventional concrete under blast load [⁷,⁸,²¹,²²]. This material model is a three-invariant model, uses three shear failure surfaces, includes damage and strain-rate effects, and has origins based on the Pseudo-Tensor model (Mat-16). The most significant advantage of this material model is the model parameter automatically generation capability based solely on the unconfined compressive strength of the concrete, mass density and passion ratio [⁷]. The other advantage of the material model is that users can easily modify the generated material parameters [²³]. In the present study, due to the accuracy and robustness, Mat-72-Rel3 is applied to model the concrete material with consideration of high strain ratio effect, plastic and shear failure surface.

Mat-72-Rel3 model only includes three independent parameters such as initial yield surface, maximum failure surface, and residual surface and can precisely simulate the dynamic behavior of concrete under impact or blast load. The failure surface of this material model is given by Eq. (1):

(1)S_i(p)=a_{\mathrm{0}i}+{\displaystyle \frac{p}{a_{\mathrm{1}i}+a_{\mathrm{2}i}\cdot p}},

where a_{\mathrm{0}i}_,a_{\mathrm{1}i}, and a_{\mathrm{2}i} are the failure surfaces parameters, S_i is theimode of failure surface, and p=({\sigma }_1+{\sigma }_2+{\sigma }_3)/3is the pressure with parameters of the first, second, and third principal stresses for {\sigma }_1, {\sigma }_2, and{\sigma }_3.

The hardening parameters of concrete incorporated with maximum failure surface and compressive strength f_c^{\text{'}}are expressed in Eq. (2).

(2)\mathrm{\Delta }\sigma /f_{\mathrm{c}}^{\text{'}}=a_{\mathrm{0}}+{\displaystyle \frac{p/f_{\mathrm{c}}^{\text{'}}}{a_{\mathrm{1}}+a_{\mathrm{2}}\cdot p/f_{\mathrm{c}}^{\text{'}}}},

where \mathrm{\Delta }\sigma =\sqrt{\mathrm{3}{J}_{\mathrm{2}}} is the deviatoric stress of failure surface, s_{\mathrm{1}}, s_{\mathrm{2}}, ands_{\mathrm{3}}are the first, second and third principal deviatoric stresses, J_{\mathrm{2}}=(s_{\mathrm{1}}+s_{\mathrm{2}}+s_{\mathrm{3}})/\mathrm{2} is the second invariant of the deviatoric stress tensor.

Reinforcement bar

For the RB in the FCP, the material model of Mat-Plastic-Kinematic (Mat-003) is adopted to simulate the dynamic property of the RB. This model is suited to model isotropic and kinematic hardening plasticity with the option of including strain rate effects. The material of Mat-003 is a very cost-effective model for simulating beam, shell, and solid elements, and the failure of the model is depended on the plastic strain, strain rate, and stress-strain curve of the RB.

Equation of state

The equation of state for the Mat-73Rel3 model is linear in internal energy. Pressure p is given in Eq. (3):

(3)p=c({\epsilon }_{\mathrm{V}})+\gamma T({\epsilon }_{\mathrm{V}})E,

where g is the specific heat ratio, and E is the initial internal energy. The volumetric strain {\epsilon }_v is given by the natural logarithm of the relative volume V.

Moreover, the tensile failure of volume will appear on the condition of tension stress reaches the hydrostatic tension boundary. The deviatoric stress tensor is demonstrated by the three-curve model.

Blast load

The blast load is simulated with the *LOAD_BLAST model in LS-DYNA. The model is derived from the blast test and developed by Glenn and Kenneth [²⁴]. The model can save the computation cost by avoiding the detailed modeling of explosive charge and propagation of shock waves in the air. Moreover, this model does not have the function of studying the effects of the shapes of explosive charges on the response of structures under blast load. The reliability of the model in simulating blast load on structures has been proven and widely used in numerical simulations of structural responses to blast loads [²⁵,²⁶].

Strain rate effect

As we all know, the high strain rate can improve the compressive and tensile strength of these materials with respect to the average strain rate. Thus, in order to reliability simulate the blast response of structures under impact loads, it is necessary to consider the strain rate effect. Furthermore, concrete and steel are strain rate sensitive materials, the effect of strain rate should be taken into consideration by employing a vital index of Dynamic Increase Factor (DIF) (seen in Eqs. (4), (5), and (6)) and incorporating into the material model.

For compressive strength of concrete [²⁷]:

(4)\mathrm{CDIF=}{\displaystyle \frac{f_c}{f_{\mathrm{cs}}}}=\{\begin{array}{l}{\left({\displaystyle \frac{\dot{\epsilon }}{{\dot{\epsilon }}_s}}\right)}^{\mathrm{1.026}\alpha },\hspace{0.17em}\hspace{0.17em}\mathrm{for} \dot{\epsilon }\le \mathrm{30}{\mathrm{s}}^{-\mathrm{1}},\\ \gamma {\left({\displaystyle \frac{\dot{\epsilon }}{{\dot{\epsilon }}_s}}\right)}^{\mathrm{1}/\mathrm{3}},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{for} \dot{\epsilon }>\hspace{0.17em}\mathrm{30}{\mathrm{s}}^{-\mathrm{1}},\end{array}

wheref_{\mathrm{cs}}andf_care the compressive strength of static load at {\dot{\epsilon }}_sand the compressive strength of concrete of dynamic load at\dot{\epsilon }, respectively. The variable \dot{\epsilon }is strain rate with the value from 30 × 10^-6 to 300 s^-1. In addition, {\dot{\epsilon }}_sis the static strain rate with the value of 30 × 10^-6 s^-1, and the other related formula is \log \gamma =\mathrm{6.156}\alpha -\mathrm{2} and\alpha ={(\mathrm{5}+\mathrm{9}{f}_{cs}/\mathrm{10})}^{-\mathrm{1}}.

For tensile strength of concrete [²⁷]:

(5)\mathrm{TDIF=}{\displaystyle \frac{f_t}{f_{\mathrm{ts}}}}=\{\begin{array}{l}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\left({\displaystyle \frac{\dot{\epsilon }}{{\dot{\epsilon }}_s}}\right)}^{\delta },\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{for}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\dot{\epsilon }\le \mathrm{1}\hspace{0.17em}{\mathrm{s}}^{-\mathrm{1}},\\ \beta {\left({\displaystyle \frac{\dot{\epsilon }}{{\dot{\epsilon }}_s}}\right)}^{\mathrm{1}/\mathrm{3}},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{for}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\dot{\epsilon }>\mathrm{1}\hspace{0.17em}{\mathrm{s}}^{-\mathrm{1}},\end{array}

where f_t and f_{\mathrm{ts}}are the dynamical tensile strength of concrete at\dot{\epsilon }and the tensile strength of static load at{\dot{\epsilon }}_s. \dot{\epsilon }is the strain rate with the value from 10^-6 to 160 s^-1, {\dot{\epsilon }}_sis the static strain rate with the value of 10^-6 s^-1, \log \beta =\mathrm{6}\delta -\mathrm{2}and\delta \mathrm{=1/}{(\mathrm{1}+\mathrm{8}{f}_{\mathrm{cs}}/\mathrm{10})}^{-\mathrm{1}}.

For RB [²⁷]:

(6)\mathrm{DIF=}\hspace{0.17em}{\left({\displaystyle \frac{\dot{\epsilon }}{{\mathrm{10}}^{-\mathrm{4}}}}\right)}^{\alpha },

where the yield strength is expressed as\alpha ={\alpha }_{f_{\mathrm{y}}}\mathrm{=0}\mathrm{.074-0}\mathrm{.04}{f}_{\mathrm{y}}\mathrm{/60}, and the equation of ultimate stress is obtained by \alpha ={\alpha }_{f_u}\mathrm{=0}\mathrm{.019-0}\mathrm{.009}{f}_y\mathrm{/60}.

Erosion criterion

The function of Add-Erosion is adopted in LS-DYNA to describe the erosion characteristic of concrete or steel under high strain rate load. In the present model, the erosion criterion of concrete material is defined as the effective plastic strain, which has been proven that can adequately model the spalling damage of concrete under blast load. Effective plastic strain can be described in Eq. (7):

(7){\overline{\epsilon }}_{\mathrm{p}}={{\displaystyle {\int }_{\mathrm{0}}^t\left({\displaystyle \frac{\mathrm{2}}{\mathrm{3}}}{D}_{ij}^{\mathrm{p}}{D}_{ij}^{\mathrm{p}}\right)}}^{\mathrm{1}/\mathrm{2}}\mathrm{d}t,

where{\overline{\epsilon }}_{\mathrm{p}}is the effective plastic strain and D_{ij}^{\mathrm{p}} is the plastic component rate of deformation.

Numerical model validation

Experimental setup

An experiment of RC slabs performed by Wang et al. [²⁰] is adapted to verify the precision and reliability of FEM for simulating the dynamic behavior and damage characteristic of FCPs under blast load. The detailed information and layout of the RC slab experiment were shown in ref.[20].

In the test, the geometrical dimensions of the RC slab were 1000 mm × 1000 mm × 40 mm (length × width × thickness). The RC slab was shored by the steel support, the opposite two sides of the RC slabs were fixed with the support and the last two edges were free. The diameter of the RB of RC slab was 6 mm with the space of each bar of 75 mm and the reinforcement ratio r of 1.43%. The explosion charge of cylindrical TNT and standoff distance were 0.46 kg and 400 mm, respectively. Moreover, the compressive strength, tensile strength, and elastic modulus of concrete were 39.6 MPa, 4.3 MPa, and 28.2 GPa, respectively. The RB’s yield strength and elastic modulus are 600 MPa and 200 GPa.

Mesh convergence

In the validation of the FEM, solid element and beam elements are applied to mesh the materials of concrete and RB, respectively. Then, mesh convergence analyses are performed to obtain an acceptable mesh size. We take the specimen FCP1-1 (seen in Table 1) as an example for calculating the mid-span displacement and obtain the acceptable mesh size. Figure 1 shows the calculated errors of displacement on mid-span slab are 14.7%, 14.7%, and 5.31% with the element sizes of 5.0, 2.5, 1.0, and 0.5 cm. Finally, we adopt an element size of 1.0 cm to mesh concrete and RB elements, and we also assume that the concrete and RB have a strong bond without any slips.

Experimental and numerical model comparison

Figure 2 plots the results of damaged contour and crack distribution both in numerical simulation and test [⁴,²⁰]. From Fig. 2(a), there are no cracks or compressive damage on the top surface of the RC slab, the damage of numerical simulation is well captured and shown in Fig. 2(b). It can be seen from Figs. 2(c) and 2(d) that some straight cracks exist in the middle bottom surface of the RC slab, which indicates that the FEM can effectively simulate the cracks.

The values of maximum displacement at the mid-span slab of FEM and experimental specimen are 10.8 and 10.1 mm, respectively, are shown in Fig. 3. While the calculated error of experimental and numerical results is nearly 6.9%. In summary, the numerical model shows a good agreement with the experimental results for the damage mode and deformation. Thus, this model has quite robust and accurate capacities in calculating the blast response of RC slabs under the explosion.

Finite element simulation

Description of FCPs

We establish a numerical model of FCP and analyze the blast resistance, the dimension and configuration of FCPs are satisfied with the code of precast concrete interior shear wall panels 15G365-2 [²⁸]. The geometry model and 3D numerical model of FCP are demonstrated in Figs. 4 and 5, respectively. The height and width of FCP are 180 and 264 cm with a thickness of 20 cm, the horizontal RB extends 20 cm on both sides. The detailed information of the FCP NQ-1828 is described in the code of precast concrete interior shear wall panels. In addition, TNT charge is located above the middle point of FCPs at the height of 60 cm. Four groups with 17 FCPs are conducted to simulate the blast performance numerically. The parameters of FCP specimens are listed in Table 1.

The solid element and beam element are used to mesh the concrete and RBs with the element size of 1.0 cm. The concrete and RB have a strong bond without any slips. At last, the total number of the elements is 40000 and 20000 for concrete and RB.

Blast load

In this study, the blast load is simulated by the Load-Blast model, which derived from blast tests by Glenn and Kenneth [²⁴]. This model can effectively save calculation time without simulating the effect of explosive charge and propagation of shock waves in the air. What is more, the reliability of this algorithm in studying blast response of structures has been validated and applied in Refs. [⁷,²⁵,²⁶].

Boundary conditions

In the present study, two opposite short sides of the FCP specimen are fixed, and the other long sides are assumed free. The influences of slab thickness, scaled distance, reinforcement ratio, and compressive strength of concrete are performed to assess the blast performance of FCPs. Moreover, the reinforcement ratio and the concrete grade are also demonstrated in Table 1, the detailed material model of concrete and steel are shown in Tables 2 and 3.

Parametric study and comparison

Damage modes

The blast resistance and damage characteristic of FCPs is studied under close blast loads, four different TNT charges of 1.0, 2.0, 3.0, and 4.0 kg detonated by electronic detonator above the mid-span top of FCPs are taken into consideration. Figure 6 shows the failure modes and effective strain contour of FCPs.

As shown in Fig. 6(a), some small narrow cracks occur in the middle bottom surface of FCPs and few penetrated cracks are observed in the horizontal direction of the panel for a TNT charge of 1.0 kg. Simultaneously, it can be seen that the FCP had small flexural deformation at the mid-span. The number of cracks and damage distribution increase naturally and turned worse with the weight of TNT increases. Figure 6(b) shows the distribution of cracks and penetration damage at the front surface of FCP for the TNT charge of 2.0 kg. It is observed that the circular cracks and penetration damage first occur in the front surface of FCP. Furthermore, the penetrated cracks increase, and concrete material spalls seriously in the bottom surface of FCP with flexural failure. We also find that the penetration damage area increases naturally with more considerable deformation at the bottom surface of FCP for the TNT charge of 3.0 kg, as shown in Fig. 6(c). The failure mode of FCP is a flexural shear failure.

Figure 6(d) shows the damage mode of FCP with the explosion of 4.0 kg TNT charge. It is observed that the area of damage failure is the biggest compared with the other three cases of the TNT charges. There are many penetrated cracks in the front surface of FCP, and the cracks are distributed in the middle area intensively. Simultaneously, the spallation area of concrete is also bigger than the other cases. The damage mode of the FCP is categorized as a shear failure.

As above mentioned, it is found that the TNT charge has a worse influence on the damage modes of FCPs. The damage modes of FCPs are changed through flexural deformation, flexural failure, and flexural shear failure to shear failure with the TNT charges increasing, respectively.

Effects of explosive charges

The influences of TNT charges on the blast resistance of FCP are numerically investigated, three TNT charges (i.e., 2.0, 2.5, and 3.0 kg) are located above the middle point of FCPs. Meanwhile, the corresponding scaled distances are 4.76, 4.42, and 4.16 m/kg^1/3, respectively, the results of the effects of explosive charges are listed in Table 4 in detail.

The results of the displacement time history of FCP under various weights of explosives are plotted in Fig. 7. It is indicated that displacements exhibit the trend of the harmonic wave with the time increasing. The reason is that the blast wave first as a compressive wave impacts the panel and then turns to be a tensile wave when it reaches the front surface of the panel, which makes the panel vibration for a long time. It is observed that the vibration periods of FCPs increase with the explosive charge increasing.

The peak displacement at the top midpoint surface of FCP is plotted in Fig. 8 under different weights of explosives. It is indicated that the peak displacement increases with explosive charges increasing, and the peak displacements are 0.615, 0.759, and 0.952 cm for the explosive charges of 2.0, 2.5, and 3.0 kg, respectively. In short, the weight of the explosive takes a significant influence on the blast behavior of the FCPs. In other words, the damage of FCP can be aggravated by increasing the weight of explosive charges.

Effects of the thickness

Generally, the thickness of the panel has a significant effect on the blast resistance of structures. In this section, three types of thicknesses with 200, 220, and 250 mm are considered and shown in Tables 1 and 5.

Figures 9 and 10 plot the displacement time history and the variation of peak displacement at the mid-span top surface. It is observed from Figs. 9 and 10 that the displacement at the mid-span top of FCP decreases obviously as the thickness increases. The maximum displacements at the middle surface of FCP 1-1, FCP 2-1, and FCP 2-2 are 0.615, 0.478, and 0.332 cm, respectively. The decrease ratios of different panels are 22.3% and 46.0% for panel thicknesses of 200, 220, and 250 mm, respectively. The blast and bending resistance improve distinctly with the thickness of FCP increasing of under the same weight of explosive, while the period and time of peak displacement response decrease to some extent. In short, the displacement of the thicker FCP is smaller, which can provide profound blast resistance capacity.

Figure 11 shows the damage modes of FCPs with different thicknesses. From the figure, it is indicated that the damaged area and cracks at the middle-upper and bottom surfaces of panels decrease with the thickness increasing. In these cases, FCP 1-1 has the severest actual spalling damage at the bottom surface and has the most number of cracks compared to other cases of panels. On the contrary, the upper surfaces of FCPs have fewer cracks and weak damage. Similarly, FCP 2-2 has the weakest spallation of concrete and the least cracks on the opposite surface, which means the thicker thickness of the panel has, the stronger blast resistance the panel has.

It is also indicated that the specimen of FCP 2-2 has a bigger cross-section and larger section moment of inertia, which improves the bending resistance and capacity of shear compared with the other cases of panels. On the other hand, the reinforcement ratio of the panel decreases as the thickness of the panel increasing, which can modify the damage mode from ductile failure to brittle failure. In conclusion, the thicker FCPs have a strong capacity against deformation or impact.

Effects of reinforcement ratio

In this section, the reinforcement ratios in horizontal direction (RRH) are taken as 0.324%, 0.506%, and 0.728% to analyze the effects of reinforcement ratios in vertical direction (RRV) of 0.444%, 0.769%, and 0.968%, respectively. The scale distance of TNT charge is 4.76 cm/kg^1/3, and parameters of the numerical model are listed in Tables 6 and 7.

Effects of RRH

Figures 12 and 13 show the displacement time history curves and peak displacement at mid-span point of FCP with different RRH and same RRV under the same blast loads. As shown in Figs. 12, 13, and Table 6, the maximum displacements of FCPs are 0.615, 0.586, and 0.573 cm for the RRH of 0.324%, 0.506%, and 0.728%, respectively. It is indicated that the peak displacement decreases as the RRH increasing. The peak displacement decreases from 0.615 to 0.573 cm with RRH increasing by 125%, while the peak displacement only decreases by 6.83%.

The effective plastic strain and failure modes of FCPs with various RRH can be observed in Fig. 14. It can be found that the damaged area of FCP1-1 is the largest compared with the cases of FCP 3-1 and FCP 3-2, which has the most number of cracks in the middle bottom area with the worst concrete spalling damage on the opposite surface. However, the failure modes of FCP 1-1, FCP 3-1, and FCP 3-2 are similar. Consequently, the RRH has a positive influence on the blast resistance of FCP.

Effects of RRV

Similarly, the displacement time history curves and peak displacements at mid-span point of FCP with different RRVs are plotted in Figs. 15 and 16. From Figs. 15, 16, and Table 7, it is observed that the maximum displacements of FCPs with the RRV of 0.444%, 0.769%, and 0.967% are 0.615, 0.575, and 0.565 cm, respectively. The peak displacement of FCP decreases as the RRV increasing slowly with a constant value of RRH, while the improvement of blast resistances of FCPs is small through increasing RRV to some extent. Furthermore, the peak displacement decreases from 0.654 to 0.570 cm corresponding with an increment of RRV from 0.444% to 0.968%, respectively. Therefore, the increment of RRV can improve the anti-blast performance.

Figure 17 plots the effective plastic strain and failure modes of FCPs with various RRV under the blast load. The numerical results of FCPs with various RRV are similar to the results of those cases with various RRH. When the RRH and RRV are different, the failure modes are the same except for the numbers and distribution of cracks on the bottom surface FCPs. On the contrary, the upper surfaces of FCPs have fewer cracks and weak damage compared with the bottom surface. The reason is that the upper surface subjects to a compressive wave while the bottom subjects to tensile stress.

Theoretical analysis

The TM

Due to the FCP satisfies with thickness less than{\displaystyle \frac{\mathrm{1}}{\mathrm{20}}}\mathrm{th}of its lateral dimension and has deflection less than{\displaystyle \frac{\mathrm{1}}{\mathrm{5}}}\mathrm{th}of its thickness, thus, the FCP is a thin plate with large defections. Figure 18 shows the FCP with two opposite edges fixed and the other two edges free, which subjects to vertical uniform load q at its upper surface.

Based on the theory of shell and plate, and elastic-plastic mechanics, for this case the maximum deflection function w (x, t) should satisfy the plate equation Eq. (8) and has the following boundary conditions (seen in Eqs. (9) and (10)).

(8){\displaystyle \frac{{\partial }^{\mathrm{4}}w}{\partial x^{\mathrm{4}}}}+\mathrm{2}{\displaystyle \frac{{\partial }^{\mathrm{4}}w}{\partial x^{\mathrm{2}}\partial y^{\mathrm{2}}}}+{\displaystyle \frac{{\partial }^{\mathrm{4}}w}{\partial y^{\mathrm{4}}}}={\displaystyle \frac{q(x,y,t)}{D}},

(9)w=\mathrm{0},\mathrm{a}\mathrm{n}\mathrm{d}{\displaystyle \frac{\partial w}{\partial x}}=\mathrm{0}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{e}\mathrm{d}\mathrm{g}\mathrm{e}\mathrm{s}\mathrm{o}\mathrm{f}x=\mathrm{0}\mathrm{a}\mathrm{n}\mathrm{d}x=a,

(10){\displaystyle \frac{{\partial }^{\mathrm{2}}w}{\partial y^{\mathrm{2}}}}+\nu {\displaystyle \frac{{\partial }^{\mathrm{2}}w}{\partial x^{\mathrm{2}}}}=\mathrm{0},\mathrm{a}\mathrm{n}\mathrm{d}{\displaystyle \frac{{\partial }^{\mathrm{3}}w}{\partial y^{\mathrm{3}}}}+(\mathrm{2}-\nu ){\displaystyle \frac{{\partial }^{\mathrm{3}}w}{\partial x^{\mathrm{2}}\partial y}}=\mathrm{0}\mathrm{f}\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{e}\mathrm{d}\mathrm{g}\mathrm{e}\mathrm{s}\mathrm{o}\mathrm{f}y=0\mathrm{a}\mathrm{n}\mathrm{d}y=b,

where q(x,y) is the vertical load and D is the flexural stiffness, which is expressed as Eq. (11).

(11)D={\displaystyle \frac{E_e{h}^{\mathrm{3}}}{\mathrm{12}(\mathrm{1}-{\nu }^{\mathrm{2}})}},

where h is the thickness of the thin plate, E_e is the equivalent elastic modulus.

Based on Levy’s theory, the vertical deflection w of FCP is represented by the following Eq. (12).

(12)w(x,t)=C_{\mathrm{1}}(t)x^{\mathrm{2}}{(x-a)}^{\mathrm{2}}\mathrm{.}

We obtained the second derivative of Eqs. (13) and (14)

(13){\displaystyle \frac{{\partial }^{\mathrm{2}}w}{\partial x^{\mathrm{2}}}}=\mathrm{2}{C}_{\mathrm{1}}(\mathrm{6}{x}^{\mathrm{2}}-\mathrm{6}ax+a^{\mathrm{2}}),

(14){\displaystyle \frac{{\partial }^{\mathrm{2}}w}{\partial y^{\mathrm{2}}}}=\mathrm{2}{C}_{\mathrm{1}}(\mathrm{6}{x}^{\mathrm{2}}-\mathrm{6}ax+a^{\mathrm{2}})\mathrm{.}

The strain energy U of the thin plate in terms of the maximum deflection w is given in Eq. (15)

(15)U={\displaystyle \frac{D}{\mathrm{2}}}{\displaystyle \int {\displaystyle {\int }_A\left\{{({\nabla }^{\mathrm{2}}w)}^{\mathrm{2}}-\mathrm{2}(\mathrm{1}-\nu )\left[{\displaystyle \frac{{\partial }^{\mathrm{2}}w}{\partial x^{\mathrm{2}}}}{\displaystyle \frac{{\partial }^{\mathrm{2}}w}{\partial y^{\mathrm{2}}}}-{\left({\displaystyle \frac{{\partial }^{\mathrm{2}}w}{\partial x\partial y}}\right)}^{\mathrm{2}}\right]\right\}}}\mathrm{d}x\mathrm{d}y\mathrm{.}

According to the energy conservation, strain energy is equal to the work by the pressure q(t) and deflection, we have

(16){\displaystyle \frac{\partial U}{\partial C_m}}={\displaystyle {\iint }_{A}q(t)w_m\mathrm{d}x\mathrm{d}y}\mathrm{.}

By substituting Eqs. (13) and (14) into Eq. (15), and considering the boundary conditions, we get Eq. (17).

(17)w(x,t)={\displaystyle \frac{q(t)}{\mathrm{2}{E}_e{h}^{\mathrm{3}}}}(\mathrm{1}-{\nu }^{\mathrm{2}})x^{\mathrm{2}}{(x-a)}^{\mathrm{2}}\mathrm{.}

In this study, the FCP is subjected to blast load, which can be assumed as a uniform load. To obtain the uniform load, we use the empirical equation to get the uniform load. The pressure of blast q(t) is obtained by the empire equation of Baker [⁴,²⁹], which has been widely used in calculating the pressure attenuation process (seen in Eqs. (18) and (19)).

(18)q(t)=q_{s\mathrm{0}}\left(\mathrm{1}-{\displaystyle \frac{t}{t_{\mathrm{0}}}}\right)e^{\frac{-\alpha t}{t_{\mathrm{0}}}},

(19)q_{\mathrm{s0}}=\{\begin{array}{l}\mathrm{20.06}{Z}^{-\mathrm{1}}+\mathrm{1.94}{Z}^{-\mathrm{2}}-\mathrm{0.04}{Z}^{-\mathrm{3}}(\mathrm{0.05}\le Z\le \mathrm{0.5}),\\ \mathrm{0.67}{Z}^{-\mathrm{1}}+\mathrm{3.01}{Z}^{-\mathrm{2}}+\mathrm{4.310.04}{Z}^{-\mathrm{3}}(\mathrm{0.5}\le Z\le \mathrm{70.9}),\end{array}

where t is the pressure wave duration time, a is the attenuation factor, q_s0(t) is the overpressure peak.

The expression for the maximum vertical deflection w of FCP is obtained by substituting Eq. (18) into Eq. (17), then we get Eq. (20).

(20)w(x,t)={\displaystyle \frac{q(t)}{\mathrm{2}{E}_e{h}^{\mathrm{3}}}}(\mathrm{1}-{\mu }^{\mathrm{2}})x^{\mathrm{2}}{(x-a)}^{\mathrm{2}}\mathrm{.}

In addition, Z is the scaled distance.

As above mentioned, the equivalent elastic modulus is assumed as one type of homogeneous material. The FCP is composed of concrete and RB. Hence, the equivalent principle of composite material is applied to obtain the equivalent elastic modulus E_e.

Figure 19 shows the equivalent principle of RC slab, the RB is assumed as a layer of steel, the section of RB is equal to the volume of RB divided the area of FCP slab. We define the thickness of RB as A_s, and the thickness of concrete is assumed as A_c.

Hence, according to the equilibrium equation of the slab, the equivalent elastic modulus E_e can be obtained by Eq. (21).

(21)E_{\mathrm{e}}=E_{\mathrm{c}}(\mathrm{1}-{\rho }_{\mathrm{s}})+E_{\mathrm{s}}{\rho }_{\mathrm{s}},

where

(22)E_{\mathrm{e}}{A}_{\mathrm{e}}=E_{\mathrm{c}}{A}_{\mathrm{c}}+E_{\mathrm{s}}{A}_{\mathrm{s}},

(23)A_{\mathrm{e}}=A_{\mathrm{c}}+A_{\mathrm{s}},

(24){\rho }_{\mathrm{s}}={\displaystyle \frac{A_{\mathrm{s}}}{A_{\mathrm{e}}}},

where E_c and E_s are the elastic modulus of concrete and RB, as listed in Eqs. (22), (23) and (24), respectively.

By substituting Eq. (21) into Eq. (20), we obtain Eq. (25).

(25)w(x,t)={\displaystyle \frac{q(t)}{\mathrm{2}[E_{\mathrm{c}}(\mathrm{1}-{\rho }_{\mathrm{s}})+E_{\mathrm{s}}{\rho }_{\mathrm{s}}]{\delta }^{\mathrm{3}}}}(\mathrm{1}-{\mu }^{\mathrm{2}})x^{\mathrm{2}}{(x-a)}^{\mathrm{2}}\hspace{0.17em}\hspace{0.17em}\mathrm{.}

DIF

In the FE model, the strain rate effects of concrete and RB are considered in the constitutive material model. For the theoretical analysis of FCP, a varying DIF in terms of strain rate is adopted to consider the strain rate effects. The DIF from Ref. [³⁰]. is used in this section.

Therefore, the DIF of concrete is given as follows:

(26)DIF={\displaystyle \frac{{\displaystyle {\int }_{\mathrm{0}}^{{\eta }_{\mathrm{0}}}{D}_{\mathrm{c}}(\dot{\epsilon }){\displaystyle \frac{k\epsilon /{\epsilon }_{\mathrm{0}}-{(\epsilon /{\epsilon }_{\mathrm{0}})}^{\mathrm{2}}}{\mathrm{1}+(k-\mathrm{2})\epsilon /{\epsilon }_{\mathrm{0}}}}\mathrm{d}\epsilon }}{{\displaystyle {\int }_{\mathrm{0}}^{{\eta }_{\mathrm{0}}}{\displaystyle \frac{k\epsilon /{\epsilon }_{\mathrm{0}}-{(\epsilon /{\epsilon }_{\mathrm{0}})}^{\mathrm{2}}}{\mathrm{1}+(k-\mathrm{2})\epsilon /{\epsilon }_{\mathrm{0}}}}\mathrm{d}\epsilon }}}\mathrm{.}

In Eq. (26), D_{\mathrm{c}}(\dot{\epsilon }) of concrete can be calculated using CEB code [²⁷].

(27)D_{\mathrm{c}}(\dot{\epsilon })=\{\begin{array}{l}{\left(\dot{\epsilon }/{\dot{\epsilon }}_{\mathrm{0}}\right)}^{\mathrm{1.026}\alpha },\left(\dot{\epsilon }\le \mathrm{30}{s}^{-\mathrm{1}}\right)\\ \beta {\left(\dot{\epsilon }/{\dot{\epsilon }}_{\mathrm{0}}\right)}^{\mathrm{1}/\mathrm{3}},\left(\dot{\epsilon }>\mathrm{30}{s}^{-\mathrm{1}}\right)\end{array}

where \dot{\epsilon }=3\mathrm{0}\times 1{\mathrm{0}}^{-6}{s}^{-1}, \beta ={\mathrm{10}}^{\mathrm{6.156}\alpha -\mathrm{2}},\alpha ={(\mathrm{5}+\mathrm{9}{f}_{cs}/\mathrm{10})}^{-\mathrm{1}}.

On this basis, by introducing Eq. (26), Eq. (27) and the adjustment coefficient W into Eq. (25), the complete equation of vertical deflection (Eq. (28)) is obtained as follows:

(28)w(x,t)={\displaystyle \frac{q(t)DIF\times \mathrm{\Omega }}{\mathrm{2}[E_{\mathrm{c}}(\mathrm{1}-{\rho }_{\mathrm{s}})+E_{\mathrm{s}}{\rho }_{\mathrm{s}}]{\delta }^{\mathrm{3}}}}(\mathrm{1}-{\mu }^{\mathrm{2}})x^{\mathrm{2}}{(x-a)}^{\mathrm{2}}\mathrm{.}

In addition, r₀ is the reinforcement ratio of the standard member of FCP.

Lastly, the maximum vertical deflection of the FCP can be approximately calculated by Eq. (29).

(29)w{(x)}_{\max }={\displaystyle \frac{q_{\max }DIF\times \mathrm{\Omega }}{\mathrm{2}[E_{\mathrm{c}}(\mathrm{1}-{\rho }_{\mathrm{s}})+E_{\mathrm{s}}{\rho }_{\mathrm{s}}]{\delta }^{\mathrm{3}}}}(\mathrm{1}-{\mu }^{\mathrm{2}})x^{\mathrm{2}}{(x-a)}^{\mathrm{2}}\mathrm{.}

Results of the TM

In this section, the results for maximum vertical deflection of the TM for FCP under blast loads compared with those from the FEM are listed in Table 8. It is indicated from Table 8 that the TM can obtain the reasonable maximum displacements compared with those results from the FEM. The results obtained from TM agree well with the values of the FEM with a maximum relative difference of less than 7.91% and 7.74%, respectively. Therefore, the proposed TM can capture the maximum displacements of FCP and can predict the peak displacement of FCP under blast loads.

Conclusions

A simplified TM and FEM considering the strain rate effects and DIF are proposed to investigate the blast resistance of FCP. First, the robust and accuracy of FCP models are validated by comparing the test results of an ordinary RC slab. Then, sensitivity analyses of parameters including panel thickness, explosive mass, RRH, and RRV are conducted to study the blast performance of FCPs. Finally, a TM of FCP under blast loads is developed to predict the maximum deflection. The numerical results show that the maximum displacement and failure modes of FCPs are affected by the parameter of RRH and RRV. Whereas, parameters of the thickness of the panel, reinforcement ratio, and explosive mass have strong impacts on the dynamic behavior of FCPs. The failure modes of FCPs are varied among bending deformation, flexural failure, flexural and shear failure, shear failure with the explosive charges increasing. It is found that the thickness of FCP can modify the failure modes from ductile failure to brittle failure. The theoretical results have good agreement with the values of FEM with maximum relative errors of less than 7.91% and 7.74%, respectively. The TM can predict the maximum displacement of FCP under blast load reasonably.

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