Concrete constitutive models for low velocity impact simulations

International Journal of Impact Engineering 132 (2019) 103329

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International Journal of Impact Engineering journal homepage: www.elsevier.com/locate/ijimpeng

Concrete constitutive models for low velocity impact simulations Dikshant Saini, Behrouz Shafei

⁎

T

Department of Civil, Construction, and Environmental Engineering, Iowa State University, Ames, IA 50011, United States

A R T I C LE I N FO

A B S T R A C T

Keywords: Concrete Constitutive models Impact loads Structural response

Concrete structures are commonly exposed to low velocity impact loads originating from windborne/waterborne debris, vehicle/vessel collision, and rock fall. For the performance assessment of concrete structures under such loads, several constitutive models have been developed to date. To compare the accuracy of the available models for practical applications, the current study evaluates four constitutive models, i.e., continuous surface cap model (CSCM), elasto-plastic damage cap (EPDC) model, Karagozian and Case concrete (KCC) model, and Winfrith concrete model. For this purpose, the constitutive models are first examined at the material level through single element simulations under basic stress paths, such as uniaxial compression and tension, as well as triaxial compression. A range of measures, such as post-peak softening, shear dilation, and confinement effect, are extracted and compared. The investigation is then extended to understand how the concrete constitutive models perform at the structure level. This is achieved by replicating drop hammer tests on reinforced concrete (RC) and concrete filled steel tube (CFST) beams. Investigation of these two structural categories provides a unique opportunity to further evaluate the accuracy of the concrete constitutive models in interaction with the most common reinforcement details. To achieve this goal, the impact responses of RC and CFST beams are compared with full-scale experimental test data. Upon understanding the capabilities of each constitutive model in predicting the structural behavior and damage, the most important modeling parameters are examined. The outcome of this study facilitates the selection and use of concrete constitutive models for the design and assessment of concrete structures subjected to various low velocity impact loads.

1. Introduction Concrete structures are used in a wide variety of civil infrastructure applications, such as buildings and bridges. The structures in service are commonly subjected to a range of extreme loading events, including low velocity impact loads (i.e., with strain rates between 10 and 100 s−1 [1]), which can originate from windborne/waterborne debris, vehicle/vessel collision, or rock fall. To understand the response of concrete structures under impact, full-scale experiments often provide the most reliable predictions. Such tests, however, require significant time, effort, and investment, which unavoidably limit the number and scope of investigations, especially in the destructive range. As an alternative, numerical studies have received growing attention, owing to the current computational power. Despite the availability of simulation capacity, the accuracy of numerical studies greatly depends on the capability of material models. For concrete structures subjected to impact loads, a number of constitutive models have been developed to date. However, there are questions concerning how the accuracy of the available models are compared for plain and reinforced concrete structures. This has motivated the current study to provide a holistic ⁎

comparison at both material and structure levels. The available concrete constitutive models can be broadly considered in three categories, depending on how plastic strains are calculated. The models in the first category calculate plastic strain increments using their associated flow and can capture dilatancy [2]. Among the examples are Mohr Coulomb, continuous surface cap model (CSCM) [3,4], and elasto-plastic damage cap (EPDC) model [5,6]. The second category of models use the Prandtl-Reuss flow theory to calculate plastic strain increments. Equations of state are required to obtain volumetric plastic strains, which are independent of the incremental flow rule. Therefore, shear dilation cannot be captured, as the shear and volumetric behavior are decoupled. Among the examples are Johnson Holmquist concrete [7], Karagozian and Case concrete (KCC) [8,9], and Winfrith concrete model [10,11]. In the third category of models, nonassociated flow is employed to calculate plastic strain increments. The plastic damage model falls in this category [12,13]. Further to finiteelement (FE) models, mesh free methods [14,15] are proven to be efficient for solving the dynamic problems involving large deformations and high strain rate effects, such as hail impact [16]. Among the existing studies, Rabczuk and Belytschko [17,18] developed a simple and

Corresponding author. E-mail addresses: [email protected] (D. Saini), [email protected] (B. Shafei).

https://doi.org/10.1016/j.ijimpeng.2019.103329 Received 11 May 2019; Received in revised form 19 June 2019; Accepted 21 June 2019 Available online 22 June 2019 0734-743X/ © 2019 Elsevier Ltd. All rights reserved.

International Journal of Impact Engineering 132 (2019) 103329

D. Saini and B. Shafei

bridge piers [22]. In a separate study, Auyeung et al. [30] proposed a damage ratio index for the performance-based design of RC bridge piers subjected to vehicle collision. Utilizing the capabilities of the CSCM, the main factors that governed the failure of RC bridge piers were identified under a range of impact loads. The CSCM model is based on an elasto-plastic damage model that incorporates the strain rate effect. There are six general categories of formulations that form this model: elastic update, plastic update, yield surface definition, damage, strain rate effect, and kinematic hardening. Among the main features of this model are a three stress invariant shear surface with translation for pre-peak hardening, an expanding/contracting hardening cap, damage based softening with erosion, and strain rate effect. The model is assumed to be isotropic. The yield surface is formulated in terms of three stress invariants, i.e., the first invariant of the stress tensor, I1, the second invariant of the deviatoric stress tensor, J2, and the third invariant of the deviatoric stress tensor, J3. The invariants are defined in terms of the deviatoric stress tensor, i.e., Sij, Sjk, Ski, and the mean pressure, P, described as:

robust method, in which crack is captured with a set of particles. For cracking particles, this method employs Eulerian kernels instead of Lagrangian kernels. This helps reduce the computational effort, while maintaining the expected accuracy. The developed numerical method can be utilized for the impact simulations that involve the penetration of an object into a concrete structure, as explored in the current study. In the study of concrete structures subjected to low velocity impact, a range of features must be captured. This includes shear dilation effect, pre-peak hardening, post-peak softening, modulus reduction, irreversible deformation, and localized damage formation and propagation. Based on a holistic review of the existing literature, there are four constitutive material models (i.e., CSCM, EPDC, KCC, and Winfrith) that offer all (or most) of the required features and have been often used for low velocity impact simulations [19–24]. The studies on the accuracy of these material models are, however, limited. This critical aspect is investigated through the current study in a systematic way. The four identified constitutive models are first compared by replicating the basic characterization tests at the material level. This illustrates the capabilities of individual models in reproducing the concrete behavior under uniaxial and multi-axial loads. The concrete constitutive models are then employed at the structure level to examine how they can predict the response of beams impacted by a drop hammer. This extension helps understand how concrete interacts with steel reinforcement. For a holistic assessment, both conventional reinforced concrete (RC) and concrete filled steel tube (CFST) beams are evaluated. The simulations necessary for the current study are performed in the LSDYNA software package [25], which is an explicit FE program used for nonlinear transient analyses of structures. The simulation results are compared with the experimental test data to identify the models that can properly capture the expected structural response. Considering that impact simulations are directly influenced by the assumptions made for the FE model, this investigation is completed with a detailed study of a variety of modeling parameters, such as hourglass coefficient, contact algorithm, and impact energy. In the absence of any established guide, this comprehensive study provides the information necessary for selecting the concrete constitutive models in such a way that the highest possible accuracy can be achieved.

I1 = 3P

(1)

J2 =

1 Sij Sij 2

(2)

J3 =

1 Sij Sjk Ski 3

(3)

The CSCM uses a complex multiplicative and three-invariant form of failure surface, Y(I1, J2, J3), proposed by Sander et al. [4].

Y(I1, J2 , J3, k ) = J2 − R2 (J3) F f2 (I1) Fc (I1, κ )

(4)

where ℜ(J3) is the Rubin three-invariant reduction factor, Ff(I1) is the shear failure surface, and Fc(I1, κ) is the hardening cap, in which κ represents the cap hardening parameter [31]. The strength of concrete depends on both pressure and shear stresses. The shear failure surface, Ff(I1), is defined along the compression meridian as:

Ff (I1) = α − λ exp−βI1 + θI1

(5)

where α, λ, β, and θ are obtained from the triaxial compression test data plotted in the meridian plane. The CSCM is capable of modeling both ductile and brittle damage through a strain-based approach. In this model, when the strain energy exceeds a predetermined threshold, damage starts to form and accumulate during the simulation time. With an automatic parameter generation feature, only unconfined compressive strength and maximum aggregate size are required as input to the CSCM.

2. Capabilities of concrete constitutive models Since concrete has a complex response to impact loads, several constitutive material models have been developed for it. The available material models have different capabilities, which influence the accuracy of predictions. This study primarily focuses on CSCM, EPDC, KCC, and Winfrith concrete models, which are often used for impact simulations.

2.2. Elasto-plastic damage cap (EPDC) model 2.1. Continuous surface cap model (CSCM) The EPDC model is the third invariant extension of the elasto-plastic cap model [25] developed by Murray and Lewis [32]. In this model, the plastic flow and damage accumulation are treated as separate processes. The plastic flow is controlled by shear stresses. This results in permanent deformations without any degradation of elastic moduli. On the other hand, the damage accumulation controls the progressive degradation of moduli (and strength) as a result of formation and propagation of microcracks and microvoids. The plastic volume change related to the collapse of the concrete's pore structure is modeled using an elliptical cap surface. This is why the model is often referred to as EPDC model. An important feature of this model is a yield surface consisting of a shear failure surface, Ff(I1), and a cap surface, Fc(I1, k), with a continuous and smooth interaction between the two of them. This model also incorporates the influence of the third deviatoric stress invariant on the shear failure of the material. The yield surface can be mathematically described using the following equation:

The CSCM was originally developed for the analysis of roadside safety structures in a project sponsored by the National Cooperative Highway Research Program (NCHRP). Based on the existing literature, the CSCM has been widely used for modeling the concrete structures subjected to progressive collapse [26,27], drop weight impact [21,28,29], and vehicle collision [22,23,30]. Yu et al. [26] developed a numerical model to predict the collapse resistance of RC beam-slab substructures under column removal scenarios. The comparison of numerical results with test data revealed that the CSCM can properly predict the distribution of cracks as well as failure modes. The CSCM has also been used in evaluating the structural response of retrofitted beams subjected to drop weights. This led to the development of a methodology to determine the impact resistance of CFRP-strengthened CFST beams [21]. Focusing on the performance of bridge structures, the CSCM was used to model the bridge piers under vehicle collision [22,23,30]. Based on rigorous impact simulations, a generalized equation was proposed to predict an equivalent static force for the design of

Y(I1, J2 , J3, k ) = J2 − R2 (I1, J3) F f2 (I1) Fc (I1, k ) 2

(6)

International Journal of Impact Engineering 132 (2019) 103329

D. Saini and B. Shafei

where I1 is the first invariant of the stress tensor defined equal to σij, J2 denotes the second invariant of the deviatoric stress tensor defined as SijSij/2, J3 is the third invariant of the deviatoric stress tensor, k is the cap hardening parameter, and ℜ(I1, J3) is the Rubin scaling function [31]. The CSCM and EPDC model use the same methodology to predict the concrete's behavior prior to the peak strength. However, they are different in the strain softening zone (i.e., post peak). To use the EPDC model, a total of 12 parameters for the shear failure surface, 5 parameters for the cap surface, and 4 parameters for the damage characteristics need to be defined. A description of these parameters along with their calibration process is provided in Jiang and Zhao [33]. The studies that have used the EPDC model for evaluating the response of concrete structures under impact loads are limited in the literature. Among the few studies available, Jiang et al. [34] explored the EPDC model to simulate the drop hammer on RC beams. The simulation results indicated that the EPDC model can closely predict the impact force, beam deflection, and crack pattern.

used to model the transition from brittle to ductile behavior under high confinement. Strain rate effects have to be incorporated into the model through a dynamic increase factor (DIF), which is translated into the expansion/ contraction of the yield strength surface. To fully define the KCC model, there are close to 50 parameters that need to be introduced, plus an equation of state that captures the pressure-volumetric strain relationship. To avoid the complexities associated with introducing all the individual parameters, the latest release of the KCC model offers an option to auto-generate the required parameters based on only the unconfined compressive strength of concrete. 2.4. Winfrith concrete model The Winfrith concrete model, which utilizes a smeared crack approach, was originally developed for finding the response of RC structures subjected to impact loads [39–42]. This model was utilized for developing a numerical model of a steel plate composite wall subjected to projectile impact [39]. Upon the validation of the numerical model, a three-step method was proposed for the design of steel plate composite walls. In another study, Thai and Kim [40] simulated the response of prestressed concrete slabs subjected to projectile impact using the Winfrith model. The study revealed how the prestressing force can be effectively introduced to increase the impact resistance of the slabs. Thai et al. [43] utilized the Winfrith model for evaluating the response of RC panels under impact loads. On the basis of simulation results, a set of formulas were obtained to predict the required thickness of the RC panels. One of the key capabilities of the Winfrith concrete model is to generate a binary output database for the details of crack, in terms of location and dimension. In this model, the volumetric and deviatoric response of concrete are decoupled. The basic plasticity model is a fourparameter model that includes invariants of the stress tensor. The Winfrith concrete model takes into consideration strain softening in tension and supports mesh regularization. The mesh regularization is performed based on either crack opening width or fracture energy. Although this constitutive model does not have an automatic parameter generation feature, most of the necessary parameters can be determined using empirical equations. The shear failure surface in this model is defined as [44]:

2.3. Karagozian and Case concrete (KCC) model The KCC model was initially developed to improve the results of physics-based FE analyses of RC walls subjected to extreme loads [35]. This model captures the key characteristics of concrete (and other cementitious materials), including strength, confinement effect, compression hardening and softening, shear dilatancy, tensile fracture and softening, biaxial response, and strain rate effect. This model has been utilized in a number of studies focusing on the structures subjected to quasi-static and impact loads [36–38]. Among them, Deng and Tuan [36] developed a FE model to investigate the impact response of CFST beams under drop weight. This led to an energy-based design procedure. In a separate effort, the KCC model was employed for the investigation of cement based composites subjected to multiple impacts [38]. The increase in the fracture energy of high-strength concrete was found to significantly improve the impact resistance of the composites under consideration. The KCC model uses three fixed deviatoric stress invariant surfaces to compute the plastic response. The formulation for the plasticity is based on a measure of damage, which is employed to develop a new yield surface at each time step of simulation. Depending on the nature of loading, initial yield surface can be hardened to a maximum strength surface or softened to a residual strength surface. The three independent strength surfaces, i.e., yield surface, Fy(p), maximum strength surface, Fm(p), and residual strength surface, Fr(p), are defined with the following equations:

Fy (p) = a0y +

p a1y + a2y p

p Fm (p) = a0 + a1 + a2 p Fr (p) =

p a1f + a2f p

Y (I1, J2 , J3) = aJ2 + λ J2 + bI1 − 1

(12)

where λ is determined as a function of cos 3θ (0 ≤ θ ≤ π/3):

⎧ λ=

(7) (8)

⎨ K cos π − ⎡ 1 ⎣3 ⎩

cos 3θ =

(9)

1

K1 cos ⎡ 3 cos−1 (K2 cos 3θ) ⎤ ⎣ ⎦ 1 3

cos−1 (K2

for cos 3θ ≥ 0

cos 3θ) ⎤ for cos 3θ ≤ 0 ⎦

3 3 J3 2 J23/2

(13)

(14)

where K1 and K2 are the size and shape factor used to define the trace of the shear failure surface in the deviatoric plane, respectively. Parameters a, b, K1, and K2 are a function of tensile to compressive strength ratio and can be determined from the experimental tests. The strain rate effect in the Winfrith concrete model is automatically considered according to CEB [45].

where a0y, a1y, a2y, a0, a1, a2, a1f, and a2f are the strength surface parameters, and p is the pressure. The KCC model defines the failure surface with an interpolation between the material's maximum strength surface and yield strength surface (or residual strength surface). The following equations are used for this purpose:

F (I1, J2 , J3) = [η (λ )(Fm (p) − Fy (p)) + Fy (p)] r (J3) for λ ≤ λm

(10)

3. Assessment of constitutive models at material level

F (I1, J2 , J3) = [η (λ )(Fm (p) − Fr (p)) + Fr (p)] r (J3) for λ > λm

(11)

The comparison of constitutive models is first made at the material level by subjecting a single cubic element of concrete to various static loading scenarios, such as uniaxial compression, uniaxial tension, and triaxial compression. In this set of simulations, the boundary conditions are defined such that both compressive and tensile performance of concrete is captured (Fig. 1). The uniaxial compression and tension are applied to the top surface using a displacement boundary condition. As

where λ is the internal damage parameter, which is a function of J2 and several other parameters. η(λ) is a function of parameter λ with the values of η (0) = 0 , η (λm) = 1, and η (λ ≥ λm) = 0 . This indicates that the failure surface initiates at the yield strength surface, reaches the maximum strength surface as λ approaches λm, and drops to the residual strength surface as λ further increases beyond λm. r(J3) is a scale factor 3

International Journal of Impact Engineering 132 (2019) 103329

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Fig. 1. Single-element model setup to investigate the response of concrete constitutive models to various loading scenarios.

for triaxial compression, a confining pressure is applied to all the six sides. Under the applied pressure, a displacement boundary condition is then used for performing the simulations. The unconfined compressive strength of concrete is assumed 42 MPa and its maximum aggregate size is 10 mm. The size of the single element is 50 mm in all the three perpendicular directions. 3.1. Uniaxial loads The stress strain relationship for the single element setup under uniaxial compression is obtained from the simulations performed on all the four constitutive models. It can be seen in Fig. 2(a) that the peak of the stress strain curves is equal to the unconfined compressive strength of concrete. This confirms that all the models capture the maximum compressive stress well. It is noted that the CSCM and KCC models predict the post-peak response with almost no strain hardening. On the other hand, the EPDC model demonstrates a linear softening part. The Winfrith model predicts a perfectly plastic response in compression, which is not realistic for concrete. To investigate the shear dilation effect, the axial stress is plotted with the volumetric strain, as shown in Fig. 2(b). Shear dilation is critical, especially where the confinement effect must be taken into consideration. This includes the RC columns, in which the force needed in the transverse reinforcement for confining the core concrete is developed due to dilation. The volumetric strain is positive for compression and negative for tension. The CSCM, EPDC, and KCC models are found to include shear dilation, but the Winfrith model is not. Fig. 3 illustrates the stress strain relationship of the single concrete element under uniaxial tension. The peak values represent the tensile strength of the concrete. It should be noted that the tensile strength is given as an input to the KCC and Winfrith models. Comparing to the tensile strength calculated based on the ACI 318 [46], i.e., 2.8 MPa, the CSCM model predicts a slightly higher and the EPDC model predicts a slightly lower value. The percentage of deviation in neither one goes above 2.7%. As for post-peak response, the CSCM and EPDC models are able to provide the softening behavior under tension. The softening part for the KCC and Winfrith models is, however, linear, noting that the drop in the Winfrith model is unrealistically slow.

Fig. 2. Stress strain relationship for the single concrete element under uniaxial compression.

triaxial compression tests on the single element setup using the boundary conditions illustrated in Fig. 1(c). Fig. 4 shows the stress strain relationship for different constitutive models under a triaxial compression of 5, 10, and 20 MPa. While the CSCM, EPDC, and KCC models are found to be stable under different confinement pressures, the KCC model is determined to be the only model that captures the brittle to ductile transition that was anticipated after increasing the confinement pressure. On the other hand, the Winfrith model does not

3.2. Triaxial loads Upon completing the investigation under uniaxial loads, the performance of the four constitutive models is compared by performing 4

International Journal of Impact Engineering 132 (2019) 103329

D. Saini and B. Shafei

Fig. 3. Stress strain relationship for the single concrete element under uniaxial tension.

provide any softening in compression. Since the Winfrith model is not capable of modeling shear dilation, the confinement effect needs to be applied through an external pressure. The model, however, is unable to provide a stable response under such pressure, as reflected in Fig. 4. While the single element analysis is helpful to understand the basic behavior of four concrete constitutive models under various stress paths, further structural response details can be extracted from the simulations that replicate the actual material test setups. For this purpose, triaxial compression tests are performed on a solid cylinder that has a diameter of 152.4 mm and height of 304.8 mm. To make direct comparisons possible, the concrete material properties are assumed the same as those used in the single element simulations. Fig. 5 shows the geometry and boundary conditions used for the triaxial compression simulations performed using the four constitutive models. The confinement effect is considered by applying a pressure on the lateral surface of the cylinder. Fig. 6 shows the stress strain relationship for the triaxial test under no, 5, 10, and 15 MPa confinement pressures. For comparison purposes, the analytical relationship extracted through the stress strain model proposed by Samani and Attard [47] has been included in this figure. In general, the KCC and Winfrith models slightly overestimate the peak strength, whereas the CSCM and EPDC models slightly underestimate the peak strength under various confinement pressures. Small instabilities are observed in the triaxial compression test results using the EPDC model. The CSCM, EPDC, and KCC models are found to capture the brittle to ductile transition with increasing the confinement effect. The results indicate that the KCC model provides the closest predictions to the analytical values obtained for the peak strength, post peak softening, and brittle to ductile transition under various confinement pressures.

Fig. 4. Stress strain relationship for the single concrete element under the triaxial test setup with a confinement pressure of (a) 5 MPa, (b) 10 MPa, and (c) 20 MPa.

4. Assessment of constitutive models at structure level: RC beams

During the original experiments, the drop hammer was released from four different heights of 0.15, 0.30, 0.60, and 1.20 m. To save the computational time, the drop hammer sphere has been allowed to fall from 1 mm above the top surface of the beam with an initial velocity adjusted to capture the expected impact velocity. The contact interactions between the drop hammer and beam have been modeled with both automatic surface to surface and eroding surface to surface contact algorithms. In the selected contact algorithms, the coefficients for dynamic and static friction have been assumed equal to 0.3, as recommended in the literature. The sampling rate for the output data is 100 kHz. The drop hammer has been assigned a rigid material model. The modulus of elasticity has been assumed equal to that of steel, i.e., 200 GPa. The density of the drop hammer has been assigned in such a way that the sphere has a total mass of 400 kg. The following DIF formulation [50] has been employed to capture how the yield and ultimate

To extend the scope of investigations from plain concrete to the concrete structures that include steel reinforcement, a set of RC beam models are subjected to impact loads. The model details are selected similar to the experimental test setup used by Fujikake et al. [48]. The RC beam's main dimensions are 1700 mm × 250 mm × 150 mm. The models are subjected to a 90 mm radius spherical drop hammer. Fig. 7 shows a schematic sketch of the simulation setup along with reinforcement details. The beam and drop hammer have been modeled with eight-node solid elements. The steel rebars have been included with beam elements, which share their nodes with the nodes of concrete's solid elements. According to the existing studies, such as Weathersby [49], the bond strength at high strain rates is 70–100% stronger than that under static/quasi-static loading conditions. Thus, the bond is often not included in the impact simulations for the sake of computational efficiency. 5

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s−1). This formulation remains valid for conventional yield strengths (i.e., between 290 MPa and 710 MPa) and strain rates ranging from 10−4 s−1 to 225 s−1. An important feature considered in this investigation is the prediction of extent of damage. Due to difference in mathematical formulation, each model has unique capabilities in providing the damage pattern. Fig. 8 illustrates a comparison of damage experienced by the beam made of four different concrete models under a drop height of 0.3 m. It can be observed that the CSCM and EPDC models predict both vertical and diagonal shear cracks. On the other hand, the KCC model overestimates the extent of damage in the RC beam. The Winfrith model has a unique capability of mapping cracks in solid elements. This model captures both vertical and diagonal shear cracks spanning from support to mid-span. Of all the four models, the EPDC model yields the best estimate of crack pattern in the RC beam under consideration.

4.1. Effect of hourglass coefficient Hourglass modes are non-physical deformation modes that can occur due to under-integrated elements. To control hourglass energy and minimize possible side effects, an appropriate hourglass coefficient needs to be chosen. In this study, the stiffness form of hourglass control is employed and the effect of a range of hourglass coefficients, including 0.1, 0.01, and 0.001, on the performance of the four constitutive models is investigated. Figs. 9 and 10 show the structural response of the RC beam, in terms of impact force and displacement, respectively. The response time histories are obtained for a drop height of 0.3 m and compared with the experimental test data. The automatic surface to surface contact algorithm has been used for this set of simulations. Figs. 9(a) and 10(a) show the impact force and displacement time histories of the CSCM model for various hourglass coefficients. The simulation results are found to be closest to the experimental results with an hourglass coefficient of 0.01. It can be observed that the CSCM model captures the first peak very well. The recorded peak impact force

Fig. 5. Details of the cylinder used for simulating the triaxial compression test on the four constitutive models.

strengths are influenced under various strain rates:

ε˙ α DIF = ⎛ −4 ⎞ ⎝ 10 ⎠

(15)

• For the yield stress: α = α = 0.074 − 0.040 • For the ultimate stress: α = α = 0.019 − 0.009 fy

y

414

u

fy 414

where fy is the steel's yield strength (in MPa) and ε˙ is the strain rate (in

Fig. 6. Stress strain relationship for the cylinder under triaxial compressive test with a confinement pressure of (a) 0, (b) 5 MPa, (c) 10 MPa, and (d) 20 MPa. 6

International Journal of Impact Engineering 132 (2019) 103329

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Fig. 7. The details of the beam setup used in this study for investigating the performance of the four constitutive models at the structure level.

impact force and displacement time histories of the Winfrith model for various hourglass coefficients. This model is the only model that underestimates the peak impact force. However, it provides a superior performance in capturing the entire displacement time-history with all the hourglass coefficients considered in the current study. 4.2. Effect of contact algorithm The impact response obtained from a numerical simulation can vary, depending on the choice of contact algorithm. Considering that the automatic surface to surface and eroding surface to surface are the two algorithms commonly used for low velocity impact problems, it is important to investigate their effects on the response of the RC beams made with four different concrete constitutive models. Figs. 11 and 12 show the impact force and displacement response time histories with a stiffness hourglass coefficient of 0.01. In the RC beam with the CSCM model, both algorithms predict the first peak of the impact force the same. The difference between them, however, begins to emerge as the simulation continues. A review of the corresponding displacement time history indicates that the automatic surface to surface algorithm consistently predicts the displacements higher (and closer to the experimental data) than the eroding surface to surface algorithm. As for the KCC model, the highest peak impact force is recorded among all the four constitutive models. This is also reflected in the displacement time history, in which the displacements reach 15 mm. The choice of contact algorithm is found to be least influential in the EPDC and Winfrith models. While both algorithms provide similar results, the predictions made with the Winfrith model is closer to the experimental test data than those made with the EPDC model. Noting that the automatic contact algorithm offers a faster solution than the eroding contact algorithm, it is recommended to be used for impact simulations unless an extensive erosion is anticipated.

Fig. 8. Damage distribution in the RC beams after the impact test for a drop height of 0.3 m using different concrete constitutive models.

decreases with decreasing the hourglass coefficient. On the other hand, the maximum displacement increases with decreasing the hourglass coefficient. This can be attributed to the softening of the elements. The maximum displacement is underestimated by up to 13% if an hourglass coefficient of 0.01 is used. Figs. 9(b) and 10(b) show the impact force and displacement time histories of the EPDC model for various hourglass coefficients. The overall trends are similar to what reported for the CSCM model. The first peak of the impact force predicted using the EPDC model is, however, closer to the experiment than that provided by the CSCM model. The same conclusion cannot be necessarily made for the second and third peaks, as their deviations are more than those from the CSCM model. For the hourglass coefficient of 0.001, the model shows hourglass modes, which result in an increase in the recorded displacement as compared to the other two hourglass coefficients. Figs. 9(c) and 10(c) illustrate the impact force and displacement time histories of the KCC model for various hourglass coefficients. The KCC model is found to be very sensitive to the hourglass coefficient. This leads to stability issues when the hourglass coefficient is decreased. Except the first peak, the impact force time history closely matches the time history reported from the experiment, especially for the hourglass coefficients of 0.01 and 0.1. The displacement time histories are, in general, higher than that observed in the experiment. The softening effect is evident in both 0.01 and 0.001 hourglass coefficients. Figs. 9(d) and 10(d) show the

4.3. Effect of impact energy To evaluate how the constitutive models perform under different impact scenarios, a range of impact energies have been considered in the current study. To facilitate the investigations, this aspect has been reflected in the impact velocity (and the drop height associated with it), assuming that the mass of the drop hammer remains unchanged. For the four constitutive models under consideration, the simulations are carried out for 0.15, 0.30, 0.60, and 1.20 m drop heights. The impact force and displacement time histories are reported in Figs. 13 and 14. For a drop height of 0.15 m, the Winfrith model provides the closest predictions for both impact force and displacement compared to the other three constitutive models. The CSCM, EPDC, and KCC models all overestimate the first peak of the impact force with a maximum 7

International Journal of Impact Engineering 132 (2019) 103329

D. Saini and B. Shafei

Fig. 9. Effect of hourglass coefficient on the impact force experienced by the RC beam made with (a) CSCM, (b) EPDC, (c) KCC, and (d) Winfrith model.

Fig. 10. Effect of hourglass coefficient on the displacement of the RC beam made with (a) CSCM, (b) EPDC, (c) KCC, and (d) Winfrith model. 8

International Journal of Impact Engineering 132 (2019) 103329

D. Saini and B. Shafei

Fig. 11. Effect of contact algorithm on the impact force experienced by the RC beam made with (a) CSCM, (b) EPDC, (c) KCC, and (d) Winfrith model.

Fig. 12. Effect of contact algorithm on the displacement of the RC beam made with (a) CSCM, (b) EPDC, (c) KCC, and (d) Winfrith model. 9

International Journal of Impact Engineering 132 (2019) 103329

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Fig. 13. Comparison of the impact force experienced by the RC beams made with different concrete material models under a drop height of (a) 0.15 m, (b) 0.3 m, (c) 0.6 m, and (d) 1.2 m.

Fig. 14. Comparison of the displacement response of the RC beams made with different concrete material models under a drop height of (a) 0.15 m, (b) 0.3 m, (c) 0.6 m, and (d) 1.2 m. 10

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data recorded during the experiment in all the constitutive models, except the EPDC model. While the CSCM model provides the closest estimates of the impact force, the Winfrith model is the only model that underestimates the impact force. For a more detailed comparison, the maximum impact force and maximum displacement are extracted as parameters of interest for comparing the four constitutive models (Table 1). In comparison with the peak impact force of 1718 kN measured in the experiment, the CSCM, EPDC, and KCC models overestimate the peak impact force. The percentage of deviation in the CSCM model is found to be 13.7%, while the same percentage goes up to 23.4 and 20.4% for the EPDC and KCC models, respectively. On the other hand, the Winfrith model underestimates the peak impact force by 12.9%. In contrast to the peak impact force, deviation in the displacement response is not significant, expect in the EPDC model. The KCC model provides an estimate of 314 mm for the maximum displacement, which has a deviation less than 7% in comparison with 337 mm recorded during the experiment. Based on this holistic comparison, the FE models are found to predict the overall response of the CFST beams very well. Caution, however, is required in selecting the constitutive model of choice to achieve the highest possible accuracy.

deviation of 44 kN recorded when the KCC model is used. As for the mid-span displacement, the KCC and Winfrith models offer the best fit to the experimental test results. The other two models underestimate the mid-span displacement by 10 to 20%. As the impact energy increases by changing the drop height to 0.6 m, the CSCM and EPDC models are found to provide predictions close to the experiments. This can be confirmed by capturing a deviation of less than 5% for the peak impact force when the CSCM and EPDC models are used. The KCC and Winfrith models, however, do not perform well as compared to the other two models, noting that the KCC model has a 30% overestimation and the Winfrith model has a 14% underestimation of the peak impact force. All the concrete models predict the mid-span displacement reasonably well, except the KCC model. Similar observations are made when the drop height is further increased to 1.2 m. Consistent underestimation is recorded in capturing the mid-span displacement using the CSCM, EPDC, and Winfrith model, while the overestimation of the mid-span displacement is magnified in the KCC model with increase in the impact energy. 5. Assessment of constitutive models at structure level: CFST beams

6. Conclusions

The investigation is further extended to CFST beams to evaluate the ability of constitutive models in capturing the confinement effects provided by the steel tube. In CFST beams, the composite action between the concrete core and steel tube enhances the strength and ductility of concrete core, resulting in an increase in the load carrying capacity. The performance of concrete constitutive models is evaluated in the CFST beams using a drop hammer simulation setup, similar to that used for the RC beams. For this purpose, the experimental test results reported by Deng et al. [51] are utilized. The CFST beams have a support-to-support clear span of 3150 mm. Using a flat-headed cylindrical impactor that has a 150 mm diameter, the CFST beam was impacted by a drop weight of 625 kg from a height of 10.2 m. Fig. 15 shows the model details developed to replicate the experimental test setup. Fig. 16 provides the impact force time histories obtained from the FE simulation of the CFST beam using the four constitutive models. The predicted impact force is completely consistent with the response

This study provided a systematic investigation of the performance of four concrete constitutive models developed for the simulation of concrete structures subjected to low velocity impact loads. The investigations began at the material level by considering a single concrete element under uniaxial and triaxial loads. The triaxial loads were also applied to cylinder-shape setups for capturing the common material characterization tests. The simulations were then extended to the structure level, in which both embedded and confined reinforcement were examined. The effects of various input parameters, such as hourglass coefficient, contact algorithm, and impact energy, on the structural response of beam setups were investigated as well. The following key conclusions were made from the study:

• Based on the simulations performed on single concrete elements, the

Fig. 15. Details of the CFST beam setup for impact simulations using the four constitutive materials models. 11

International Journal of Impact Engineering 132 (2019) 103329

D. Saini and B. Shafei

Fig. 16. Comparison of the impact force experienced by the CFST beam made with (a) CSCM, (b) EPDC, (c) KCC, and (d) Winfrith model.

Table 1 Details of impact simulation results for the CFST beam using the four concrete constitutive models. Material Model

Max. Impact Force Result (kN) Difference (%)

Experiment CSCM EPDC KCC Winfrith

1718 1954 2120 2068 1496

•

•

– +13.7 +23.4 +20.4 −12.9

•

Max. Displacement at Mid-Span Result (mm) Difference (%) 337 310 204 314 303

– −7.9 −39.5 −6.8 −9.8

•

CSCM, EPDC, and KCC models predict elastic-plastic response with almost no hardening under compression. Among the four models, the KCC model provides the best estimates of the concrete's key response characteristics, such as post-peak softening, shear dilation, and confinement effect. The Winfrith model cannot model the postpeak softening and shear dilation effect in compression. However, it is found capable of modeling the post-peak softening in tension. From triaxial compression tests on single concrete elements, the CSCM, EPDC, and KCC models were found stable under different confinement pressures. However, only the KCC model captured the brittle to ductile transition with increasing the confinement pressure. The Winfrith model failed to capture the softening behavior in compression and was not able to provide stable response under confinement effects. Triaxial compression simulations on solid cylinders revealed that the KCC model provides the closest estimates of peak strength, post peak softening, and brittle to ductile transition under various confinement pressures, in comparison with the analytical calculations. On the other hand, the CSCM and EPDC models have a slight underestimation of the peak strength as compared to the analytical

•

results, noting that minor instabilities are observed in the EPDC model. The effects of various modeling parameters, such as hourglass control, contact algorithm, and impact energy, were studied by investigating the impact response of RC beams. The hourglass coefficient was found to have a significant effect on the performance of the concrete constitutive models. The KCC model, in particular, showed a considerable sensitivity, leading to unrealistic predictions when a low hourglass coefficient was used. For all the constitutive models, the automatic contact algorithm was able to develop appropriate results, while saving the computational time, in comparison with the eroding contact algorithm. The effect of different types of confinement was studied by investigating the impact response of both RC and CFST beam models. Despite having some deviations, the predicted maximum impact force for CFST beams did not go above 23.4% in any of the four constitutive models with the closest predictions provided by the Winfrith model. Except for the EPDC model, the consistency of results was improved when predicting the maximum displacement at the mid-span of the CFST beam models under consideration. While the computational framework presented in this study can be employed for uncertainty analysis, the current study was primarily focused on the investigation of the performance of different constitutive models, including a study of contributing input parameters. The incorporation of uncertainty of input parameters, as explored by Hamdia et al. [52], into the current study is recommended as a future step to help decide on the most optimal models and input assumptions.

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