Influence of aggregate size on shear mechanism of reinforced concrete beam subjected to impact load

Abstract

This study presents the influence of aggregate interlock, in terms of aggregate size, on shear mechanism of reinforced concrete beam under a drop-weight impact using a non-linear finite element model. Four aggregate sizes (8-, 16-, 24-, and 32-mm) were used to study the variation of the shear force at the critical section. Large shear force was recorded at the critical section for the specimens with the 32-mm aggregate size regardless of the impact mass and velocity. This is expected since larger aggregates create rougher surfaces and transfer larger shear forces than smaller ones. As the impact mass increased from 868 to 1700 kg, the critical section shear force decreased on average by 5.3%, regardless of aggregate size. Increasing the impact velocity by 41% decreased the critical section shear force by 58%, 56%, 45%, and 41.5% for the 8 mm, 16 mm, 24 mm, and 32 mm aggregate size specimens, respectively. The influence of aggregate size on the shear transfer mechanism was also studied in the presence of shear reinforcement and increased span-to-depth ratio. The difference in shear force at the critical section among the specimens with the different aggregate sizes is negligible in the presence of shear reinforcement indicating that the presence of stirrups reduced the contribution of aggregate interlock in shear resistance. Increasing the span length by 67% (from 3 to 5 m) decreased the effect of aggregate size by 95% on average.

Article highlights

• Aggregate interlock contributes to shear resistance under impact load conditions.

• Cracked sections with larger aggregate sizes transfer larger shear force along the crack surface in comparison with the ones with smaller aggregates.

• The contribution of aggregate interlock in shear resistance decreases in the presence of shear reinforcement.

1 Introduction

Structural members made of reinforced concrete (RC) are subjected to different loads based on their intended purposes. Their failure mechanisms are dependent on both natures of the loads and their orientation. RC structural members have different failure modes under static and dynamic loads of equal magnitude, orientation, and point of application. Impact load is one of the dynamic load types that could be applied to reinforced concrete structural members. It may be in the form of falling objects and concentrated loads applied at high rates [1].

Previous researches showed that when the impact velocity was low (i.e., slow loading rate), vertical cracks emerged at the point of application, indicating the flexural type of failure. These cracks later propagate outward and downward through the section of the beam. On the other hand, for high impact-velocity (i.e., fast loading rate), diagonal shear cracks developed and dispersed around the loading point rapidly indicating a high probability of shear mode of failure [1, 2]. A structural member designed to fail by flexure under static load may fail in shear under dynamic load. Based on this it was indicated that the amount of transversal reinforcement that had to be used to ensure flexural failure under dynamic load was different from that of the static load. Most design codes suggest an equivalent dynamic load factor for the design of members subjected to dynamic load [3].

The shear resistance of a reinforced concrete member is a contribution of two parts, concrete capacity $\left( {V_{c} } \right)$, and shear reinforcement (stirrup) capacity $\left( {V_{s} } \right)$, in which the former is the combined effect of concrete tensile strength (which is effective until a crack is formed), aggregate interlock, dowel action, and inclination of the compression cord (strut) [4]. Figure 1 shows these shear resisting mechanisms, where $V_{c}$ is the shear force transferred in the concrete compressive zone, N_c is the induced concrete compression force, $\tau_{ai}$ is the shear stress transferred by aggregate interlock, $\delta_{ai}$ is the normal stress at the crack, $V_{ai}$ is the shear force transferred by aggregate interlock, $V_{d}$ shear force transferred by dowel action and T_s is the tensile force in the longitudinal reinforcement.

Fig. 1

Shear transfer mechanisms in a cracked concrete section [5]

Because the strength of the hardened cement paste in most concretes is lower than the strength of the aggregate particles, cracks intersect the cement paste along the edges of the aggregate particles. Thus, aggregate particles, extending from one of the crack faces, interlock with the opposite face and resist shear displacements [6].

The bending load capacity of an RC beam under impact load can be estimated as 2.3 times its static capacity [7]. Under dynamic load, the failure mode usually changes from flexural to shear due to the result of the short period nature of the impact phenomena. The impact load may not cause shear failure but rather the high-energy loads that produce a shear-type failure mechanism [3].

The contribution of aggregate interlock is affected by the stirrup due to the confinement effect on concrete. The crack opening and the relative displacement of the two crack surfaces are the major factors in determining the contribution of aggregate interlock in shear resistance. The concrete constituents’ properties such as concrete strength, aggregate type and size, and member shear span-to-depth ratio also affect the effect of aggregate interlock.

Walraven [6] conducted in-depth research in the area of aggregate interlock modeling in the early 1980s and later on the advancement of his theories by modifying and including different parameters. He proposed a model for the shear force on the crack surface as a function of the crack width and shear slip. Gambarova and Karakoç [8] included the effect of maximum aggregate size proposing a better formulation for tangent shear modulus.

Due to its complex nature, it is not easy to directly determine the effect of aggregate interlock on shear resistance. According to Paulay and Loeber [9], about 70% of the total shear was resisted by the aggregate interlock in beams without shear reinforcement. The remaining 30% is resisted by the uncracked section in the compression zone and the dowel action of the longitudinal reinforcement. Similarly, according to the experimental study of Völgyi and Windisch [10], the contribution of aggregate interlock is dominant in the shear resisting mechanism.

There are limited research works that relate dynamic loading conditions with aggregate interlock mechanism, and thus aggregate size, in shear resistance of concrete. The main objective of this research is to investigate the contribution of aggregate interlock on the shear mechanism of reinforced concrete beams subjected to high impact-velocity load using a non-linear finite element (FE) model. The FE models are validated against experimental results obtained from other researchers.

2 Materials and methods

Nonlinear finite element (NLFE) analysis software have proven their applicability and modeling capacity through different validations against full-scale experimental test results. LS-DYNA, one of the most widely used NLFE software applicable for simulations of impact and blast, was used for this work. Experimental tests conducted by other researchers were used to validate the NLFE modeling.

2.1 Experimental specimens

De-Bo Zhao et al. [1] studied the shear behaviors of RC deep beams subjected to impact load. The authors conducted 10 large-scale drop-weight tests. The test specimens were divided into three series, namely B, C, and D, based on their span length and the shear reinforcement ratio (spacing). The specimens’ details are given in Fig. 2. For main reinforcement, ultimate strength of 620.2 MPa and yield strength of 495.5 MPa were considered whereas ultimate and yield strengths for the transverse (shear) reinforcement were 550.4 and 344.7 MPa, respectively. The specimens are designated by series name followed by impact weight and impact velocity.

Fig. 2

Specimens’ details

2.2 Specimens for numerical study

From the experimental specimens, B-1700-4.6, C-1700-4.6, and D-1700-4.6 were used for model validation. The three test specimens were selected based on their shear span-to-depth ratio and shear reinforcement ratios. Once the finite element model was verified against the experimental result, the specimens were used for further parametric study. As it is required to determine the influence of aggregate interlock in the shear mechanism, shear reinforcements were ignored in all cases except for the specimens in which the shear reinforcement was a parameter to be studied.

The influence of aggregate interlock was studied under variable impact mass and impact velocity. Four types of impact masses and velocities were adapted from the experimental specimens. To reduce computational time and resources, the impact heights were changed into an equivalent impact velocity indirectly applying kinetic energy to the drop weight. Using the conservation of energy, $v = \sqrt {2gh}$, the equivalent velocities for drop heights of 3 m, 4 m, 5 m, and 6 m were 7.7 m/s, 8.86 m/s, 9.9 m/s, and 10.85 m/s, respectively. The description of the simulation specimens was given in Table 1. Each specimen was simulated using four aggregate sizes of 8 mm, 16 mm, 24 mm, and 32 mm as an input in the concrete model in LS-DYNA.

Table 1 Specimens for parametric study

2.3 Finite element modelling

The FE models of the specimens were created in the NLFE software LS-DYNA, a commercial finite element software for non-linear simulations developed by Livermore Software Technology Corporation [LSTC].

The model creation, the input process, and the post-processing were performed in Ls-PrePost.¹ The analysis was performed using LS-DYNA solver. The material properties, element types, and other input parameters were based on the three volumes of the LS-DYNA keyword user’s manual and LS-DYNA Theory Manual [11]. The model for a typical specimen was given in Fig. 3.

Fig. 3

LS-DYNA Model Orientation

From the various concrete constitutive models in LS-DYNA, the Continuous Surface Cap (CSC) Model—MAT159 was used to model the concrete. The CSC model captures the basic concrete behaviors even though it works only under low confinement phenomena. It allows automatic parameter generation based on the unconfined compressive strength of the concrete and maximum aggregate size [12]. MAT_PLASTIC_KINEMATIC (Material type 03) was used to model the longitudinal reinforcement. This material model was suitable for modeling isotropic and hardening plasticity. To model the shear reinforcement, *MAT_PIECEWISE_LINEAR_PLASTICITY (Material type 24) was used.

The support system was reduced to a solid cylinder at the beam bottom and a solid plate at the beam top. Both the bottom and top supports and the drop weight were modeled using a rigid material (MAT_RIGID) as their responses were not needed. To be realistic, the input parameters were adopted from a typical steel material property. The reinforcement bars were modeled as a beam element. A perfect bond between the reinforcement and the concrete was assumed, and thus the reinforcements were assumed fully embedded in the concrete paste. Based on these assumptions, the interaction between the reinforcement and the concrete was taken as *CONSTRAINED_BEAM_IN_SOLID. AUTOMATIC_SURFACE_TO_SURFACE was used for the contact between the concrete beam and the supports. This contact algorism allows the contact forces to transfer from the slave to the master with no element constraint. The coefficient of fiction defined in the contact keyword between two impact masses affects the magnitude of the peak impact force. A friction coefficient of 0.2 was adopted in this work for both the static and dynamic cards based on suggestions from the literature [13].

LS-DYNA, by default, was an explicit NLFEA program. The central difference method (CDM) was used to solve the transient dynamic equilibrium equation in LS-DYNA. This default integration method was used for the simulation of the model of this work. For all specimens, a simply supported reinforced concrete beam was considered. Both the cylinder and the plate at the support were constrained against translation in the three orthogonal directions. The constraint was applied using the BOUNDARY_SPC_SET keyword on the nodes along the centerline in the transversal direction of the beam. The impact load was modeled by defining an initial velocity for the drop weight. For the model specimens of variable impact mass, the drop weight was controlled by the mass density with a constant volume.

The constant stress solid element type, known as element form 1 (element type 1) in LS-DYNA, was used to model the concrete beam. The element is under integrated constant stress, accurate and efficient element form. A mesh size of 20 mm was adapted after the sensitivity study. The use of 8-node hexahedron solid elements with one integration point requires hourglass stabilization. For this specific work, type 6 or the Belytschko-Bindeman assumed strain co-rotational form was used for the CSC concrete model. The drop-weight was modeled using the four-node tetrahedron solid element, solid type 10 in the LS-DYNA keyword, with a single integration point. It is suitable to generate solid elements with curved surfaces from geometrical shapes. Figure 4 shows the element mesh for both the beam and the drop-weight.

Fig. 4

FE Mesh of beam and drop weight specimens

3 Results and discussion

3.1 Model validation

Despite their versatile applications, the accuracy of finite element models must be supported and verified with experimental results to make sure that the model components, such as material models, boundary conditions, and contact interaction mimic the actual behavior of the system. Time histories of impact force, support reaction force, and midspan deflection were used to verify the model against the experimental work reported by De-Bo Zhao et al. [1].

Figure 5 shows the mid-span deflections of specimens B-1700-4.6 and C-1700-4.6. The simulation results agree with the experimental outputs with a slight deviation around the final duration of the simulation. At this stage, cracks had already developed and the ultimate strength of the material was exceeded dividing the beam into different components. This results in a non-continuum mechanics problem that could not be captured by the current FE model. The maximum mid-span displacements of the experiment and the finite element model were 83.04 mm and 84.4 mm, respectively for B-1700-4.6. The C-1700-4.6 specimen experienced sudden failure without rebound and the maximum mid-span deflection cannot be determined. Therefore, in terms of midspan deflection, the numerical models well captured the experimental results with acceptable accuracy. From this observation, the numerical models can be used for further parametric study.

Fig. 5

Midspan deflection: a B-1700-4.6 and b C-1700-4.6

The time history plots of the impact force and the support reaction forces were given in Figs. 6 and 7, respectively. The simulation outputs were in good agreement with the experimental results for both specimens. The peak values of the impact and the reaction forces were well represented by the CSC concrete model. The time lag between the experimental results and the simulation outputs may be because of the location of the impactor at the beginning of the simulation. In the models, the impactor was applied with an initial velocity corresponding to a specific drop height and its location was 1 mm above the top surface of the beam while the experimental setup was free-falling and the results started to be recorded at the moment the impactor hits the beam. The negative value of the support reaction forces in the early stages of the impact event shown on the experimental output plot did not happen on the simulation output for both series. This was because, in the experiment, the load cell (gauge) reads the reaction forces both at the top and bottom of the beam separately while only the bottom support reaction forces were plotted for the case of the FEM. According to the authors, the experimental results showed that B-1700-4.6 failed in flexure and the C-1700-4.6 specimen experienced shear failure.

Fig. 6

Impact force: a B-1700-4.6, b C-1700-4.6

Fig. 7

Support reaction force: a B-1700-4.6, b C-1700-4.6

The failure mode of C-1700-4.6 was observed with the element erosion future of the CSC concrete model. The elements that reached a plastic strain of 1.05 were removed from the beam near the final failure at 40 ms, and the failure modes were similar to those observed in the experimental ones, as shown in Fig. 8a.

Fig. 8

The overall failure of the C-1700-4.6 specimen, a experiment, b FEM, c Reinforcement effective stress (Von-Mises)

3.2 Influence of aggregate size on shear mechanism

Different sections were defined along the beam length between the supports to extract the shear force at different times. The *DATABASE_CROSS_SECTION_SET keyword from LS-PrePost was used to define the cross-sections at the desired location along the beam. A total of 13 sections were defined to record the shear force at any time of the simulation (Fig. 9). The influence of aggregate interlock on the amount of shear on these sections was investigated under variable mass and variable velocity. The shear force diagram and bending moment diagram of an RC beam subjected to impact load were quite different from the static loading condition. Figure 10 shows the shear force variations across the beam length for the first 10 ms.

Fig. 9

Thirteen sections at 250 mm

Fig. 10

Dynamic shear force diagram

The shear force along the beam's span was one of the major parameters influenced by the aggregate size. A shear plug was formed at the impact zone approximately 45 ̊ to the longitudinal axis and the shear critical section was a vertical section crossing the shear plug at some distance from the impact point. The time history plots of the shear force at the critical section of both the variable impact mass and impact velocities (shown in Figs. 11, 12 respectively) show that the difference among the shear force values of the four different aggregate sizes starts to increase after the peak impact force. There were also differences in the peak shear force values for four different aggregate sizes. Taking CM1 for instance, the peak critical section shear forces are 721.49 kN, 746 kN, 760 kN, and 768 kN for the specimens with 8 mm, 16 mm, 24 mm, and 32 mm aggregate size specimens, respectively.

Fig. 11

Time history plots of the critical section shear force a CM1, b CM2, c CM3, d CM4

Fig. 12

Time history plots of the critical section shear force a CV1, b CV2, c CV3 and d CV4

The time history plots of the 13 sections of the beam were converted into shear force diagrams at the different stages of the simulation time. Figure 13 shows the typical shear force distributions of the CM1 specimens at 2 and 15.5 ms of the simulation time. The influence of aggregate interlock was seen during the post-peak impact period of the simulation. This indicates that the aggregate interlock action was activated after some milliseconds of the peak impact force. As time increases, the amount of shear force on the cross-sections becomes different in magnitude for the different aggregate sizes. A significant change started to take place at the 6^th millisecond. The 32 mm aggregate size model specimens experienced larger shear force, especially for the sections away from the midspan as the shear force increases toward the support. At the beginning of the impact phenomena, i.e., from 1 up to 5 ms, the models with the different aggregate sizes experienced identical shear force distribution across the beam’s longitudinal length. At these early moments, the crack widths are too small and the influence of aggregate interlock in shear resistance is null. Once the crack has been opened, the amount of shear transferred across the crack surface depends on the roughness of the cracked section.

Fig. 13

Shear force distribution along the beam span at different times for CM1

This roughness of the surface is affected by the size of coarse aggregate used in the concrete, justifying the reason that the 32 mm aggregate size models experience a greater amount of shear force in comparison with the 8-, 16-, and 24-mm aggregate size models.

3.2.1 Variable impact mass

To investigate the influence of aggregate size on the shear mechanism under variable impact mass, it is better to take the later times to observe the differences among the four aggregate sizes. The contribution of aggregate size was observed in the post-peak-impact force region which corresponds to the time after the 6th milliseconds. Figure 14 shows the shear force diagrams of the specimens of the four impact masses each with the four aggregate sizes. The specimens with the larger aggregate sizes experienced larger shear forces in all the impact masses. The shear force diagrams in the later stages were drawn at different discrete times and the differences among the specimens of the different aggregate sizes were studied.

Fig. 14

Shear force diagrams at t = 12.5 ms for a CM1, b CM2, c CM3 and d CM4

In general, the contribution of aggregate interlock didn’t affect the magnitude of the shear force at the critical section significantly with increasing impact mass. Besides the shear variations among the different aggregate size specimens, the peak shear force at the critical section was nearly similar for all the four impact masses. The bar graph shown in Fig. 15 shows the average magnitude of shear force for the time range of 8 ms to 14.5 ms. The peak shear forces of the 32 mm aggregate size specimens, for instance, for CM1, CM2, CM3, and CM4 are in the range of 206 kN—219 kN. These differences are negligible for the large impact masses considered in the simulation. As an average, the difference between the shear force at the critical section of the 32 mm aggregate size specimen and that of the 8 mm aggregate size specimen increased from 46% for CM1 to 51% for CM4. The difference between the 32 mm and 16 mm aggregate size model specimens increased from 23.8% for CM1 to 25% for CM4. The difference between the 32 mm and 24 mm aggregate size specimens is not consistent to conclude. These differences are insignificant considering the big differences among the impact masses. The impact masses considered in this paper are large in magnitude and considering the individual impact masses, there is a significant change in the shear force at the critical section of the beam for the different aggregate sizes.

Fig. 15

Average post-peak shear force on the critical section under variable impact mass

3.2.2 Variable impact velocity

It has been said that the aggregate size, so that the aggregate interlock, affects the amount of shear force on the beam cross-section. This contribution was investigated by varying the impact velocity. The time states for the investigation are the ones taken for the variable mass. Figure 16 shows the shear force diagrams of the specimens of the four impact velocities each with the four aggregate sizes at 10.5 ms of the simulation time. The first thing observed from the plots of the shear force diagrams is that specimens with larger aggregates experienced larger shear forces regardless of the impact velocity. The other thing is that the shear force on the beam decreased with increasing the impact velocity.

Fig. 16

Shear force diagrams at t = 10.5 ms for a CV1, b CV2, c CV3, and d CV4

The plot of the average post-peak shear force at the critical section for the four different velocities is shown in the bar graph in Fig. 17. The shear force at the critical section decreased with increasing impact velocity regardless of the aggregate size. The increased impact velocity results in a sudden failure and the concrete section couldn’t resist much shear before the failure occurred. By observing the average shear force responses of the models with the four aggregate sizes under variable impact velocity at different times after the occurrence of the peak impact force, it can be said that the difference between the 32 mm and the 8 mm aggregate size model specimens increased from 36.3% for CV1 to 54.3% for CV4. The increment is from 18% for CV1 to 38.5% for CV4 and from 9.27% for CV1 to 15% for CV4 for the 16 mm and 24 mm aggregate size model specimens, respectively.

Fig. 17

Average shear force on the critical section under variable impact velocity

3.3 Influence of aggregate size: increased span-to depth ratio

The time history plots of the shear force at the critical section for the four aggregate sizes of the BM1specimens, given in Fig. 18, show that there is no significant difference between different aggregate sizes. A small difference is observed at the maximum shear force at the critical section for the four aggregate sizes. For the case of the C-series specimens, the shear forces at the critical section after 5 ms are different with significant values for the four aggregate size specimens (Fig. 11a). The post-peak values of the shear force are similar in the case of B-series specimens.

Fig. 18

Critical section shear force–time histories for BM1

The shear force distributions across the span length of the beam at two different times (3 ms and 14.5 ms) of the simulation are given in Fig. 19. The shear force diagrams show that there is no significant difference among the four aggregate size specimens as in the case of the C-series specimens. At 3 ms, all the four specimens experienced an approximately similar amount of shear force and the maximum shear force occurred at the critical section of the beam. As the time increased, the shear force experienced by the beam decreased. The maximum shear forces at different times occur at locations different from the critical section. The shear force diagrams of the C-series show that the maximum shear force occurs at the critical section of the beam for most of the simulation time. Increased span length changed this shear mechanism as the bending moment increased on the beam.

Fig. 19

Shear force diagrams for BM1 at t = 3 ms and 14.5 ms

3.4 Influence of aggregate size in the presence of shear reinforcement

The influence of aggregate size was investigated in the presence of shear reinforcement for both B and C-series specimens. The shear reinforcement ratio considered was taken from the literature [1] as 0.094%. Figure 20 shows the time history plots of the shear force at the critical section of specimens BM1 and CM1 with the four different aggregate sizes.

Fig. 20

Critical section shear force–time histories for a BM1-S and b CM1-S

The difference between the four aggregate size specimens has become insignificant for both B and C-series in the presence of shear reinforcement. Specifically, the difference is almost zero for the BM1 specimens. This is due to the combined effect of the large shear-span-to-depth ratio and the presence of shear reinforcement. The contribution of the aggregate interlock is affected both by the shear span-to-depth ratio and shear reinforcement. For the case of the CM1 specimens, an observable difference was seen at about 9 ms. As compared to the CM1 specimens without shear reinforcement, the presence of shear reinforcement eliminates the influence of aggregate interlock. Figure 21 shows the shear force diagram of the beam at the time of 2.5 ms and 11.5 ms for the BM1 specimens. The shear force along the beam span length is almost similar for the four aggregate sizes. The difference between the aggregate sizes for the post-peak shear period can be best described by the differences observed on the shear force diagram at t = 11.5 ms.

Fig. 21

Shear force diagram for BM1-S at 2.5 ms and 11.5 ms

The shear force diagrams for the CM1-S specimens at t = 2.5 ms and t = 11.5 ms are given in Fig. 22. These diagrams show that the difference in shear force among the four aggregate sizes is insignificant not only at the critical section but also at other locations along the beam span length. There is a considerable difference between these models without shear reinforcement. Even though RC beams with a smaller shear-span-to-depth ratio are susceptible to shear, the presence of shear reinforcement reduces the influence of aggregate interlock in shear resistance. One explanation for this can be related to the parameters contributing to crack. In the presence of shear reinforcements, larger crack width and smaller shear slide (relative displacement of the crack surfaces) are expected. As indicated by the modified compression field theory [14] the shear force transferred by aggregate interlock is inversely proportional to the crack width. In addition to this, most of the shear force experienced by the beam during impact is carried by the stirrups.

Fig. 22

Shear force diagram for CM1-S at 2.5 ms and 11.5 ms

4 Conclusions

One of the shear resisting methods in a cracked section of a reinforced concrete beam, aggregate interlock, is a function of aggregate size. Aggregate interlock directly relies on the aggregate size used in the cement mix matrix. The influence of aggregate size on the shear mechanism of a reinforced concrete beam was investigated in this paper. Based on the numerical model outputs, it was concluded that RC beams cast with a larger aggregate size experience larger shear force throughout the beam length including the critical section. As the impact mass increased from 868 to 1700 kg, the critical section shear force decreased on average by 5.3%, regardless of aggregate size which indicates that the influence of impact mass on the contribution of aggregate interlock is insignificant. Increased impact velocity decreased the post-peak average shear force on the critical section of the beam significantly. The critical section shear force of the 32 mm aggregate size specimen decreased from 240 kN for CV1 to 140 kN for CV4. In percentage, increasing the impact velocity by 41% decreased the critical section shear force by 58%, 56%, 45%, and 41.5% for the 8 mm, 16 mm, 24 mm, and 32 mm aggregate size specimens respectively. On the other hand, the contribution of aggregate interlock increased with increased impact velocity. Both the influence of aggregate size on the shear force distribution along the beam span and the critical section shear force decreased with increasing the span length. Increasing the span length by 67% (from 3 to 5 m) decreased the effect of aggregate size by 95% on average. The presence of shear reinforcement decreased the influence of aggregate size on the critical section shear force of the beam.