Uncertainty of concrete strength in shear and flexural behavior of beams using lattice modeling

Abstract

This paper numerically studied the effect of uncertainty and random distribution of concrete strength in beams failing in shear and flexure using lattice modeling, which is suitable for statistical analysis. The independent variables of this study included the level of strength reduction and the number of members with reduced strength. Three levels of material deficiency (i.e., 10%, 20%, 30%) were randomly introduced to 5%, 10%, 15%, and 20% of members. To provide a database and reliable results, 1000 analyses were carried out (a total of 24000 analyses) using the MATLAB software for each combination. Comparative studies were conducted for both shear- and flexure-deficit beams under four-point loading and results were compared using finite element software where relevant. Capability of lattice modeling was highlighted as an efficient tool to account for uncertainty in statistical studies. Results showed that the number of deficient members had a more significant effect on beam capacity compared to the level of strength deficiency. The scatter of random load-capacities was higher in flexure (range: 0.680–0.990) than that of shear (range: 0.795–0.996). Finally, nonlinear regression relationships were established with coefficient of correlation values (R²) above 0.90, which captured the overall load–deflection response and level of load reduction.

Graphical abstract

Keywords

lattice modeling / shear failure / flexural failure / uncertainty / deficiency / numerical simulation

1 Introduction

Concrete is a brittle construction material extensively used in various engineering practices, such as buildings, dams, bridges, etc. Once these structures fail to meet the required serviceability criteria due to aging, environmental conditions, etc., disastrous events will ensue, costing both human lives and property damage [1]. The uncertainty inherent in key material parameters, such as the compressive strength of concrete (i.e., given the multi-phase structure of concrete and the inevitable uncertainties intrinsic to casting, curing, and supervision processes, the existence of weak regions, at least on the local scale, is inevitable) calls for rigorous measures for concrete preparation and supervision during casting. Meanwhile, the formation of cracks is a trans-scale process that includes the formation of micro- and meso-scale cracks, their propagation, coalescence to macro-cracks, followed by structure failure at the final stage [2,3]. Due to its importance, numerous methods have been adopted by researchers over the last several decades to characterize the fracture behavior of concrete and better address the uncertainty inherent in different aspects of its application [4]. Some methods that use discrete methods to account for crack growth include the finite element method (FEM) [5–13], extended FEM [14], mesh-free methods [15,16]. Another branch of these methods utilizes the smeared crack approach, where crack cannot be simulated separately, including the continuum damage mechanic methods [17] and phase-field methods [18,19]. An alternative framework is founded on discontinuity models where the continuum is discretized into particles [20,21] and lattices [22]. Methods that rely on this approach include the discrete element method (DEM) [23], bonded particle model (BMP) [24], and discontinuous deformation analysis (DDA) [25].

One of the most common ways to simulate crack growth is the lattice method [26], which characterizes a 2D and/or 3D cohesive solid using 1D elements, such as springs [27], Euler-Bernoulli beams [28], and Timoshenko beams [29]. Since 1941 when Hrennikoff [30] formulated the fundamentals of this work, lattice models have gained popularity [31] as reasonable simulation results have been obtained using this approach [32].

Aydin et al. [33] adopted a 2D meso-scale lattice approach to simulate the response of various reinforced concrete (RC) structural elements. The capability of lattice models to act as tension-only materials with few input parameters was highlighted as its advantage. It was reported that the lattice approach yielded satisfactory results. However, additional research is required to further validate this simplification. In another study, Aydin et al. [34], Le and Bažant [35] proposed grid perturbation, a function of compressive strength and fracture energy to indirectly account for compressive failure as a result of tension failure. It was shown that compressive failure and strength equate to indirect tensile failure with material instability consideration (local stability loss).

Concerning the uncertainty inherent in concrete, Le and Bažant [35] argued that despite the existence of several databases for concrete strengths, such as tests on cylindrical specimens [36], shear capacity of RC beams [37], and punching shear of RC slabs [38], they are uncoordinated and have been carried out at different laboratories, with different test setups, equipment, etc., on individual specimens. As a result, the randomness adds to the complexity of achieving a reliability-based structural design for which the structural failure probability should be less than per lifetime [39] (this probability is in the same order of magnitude as death by lightning or a fallen tree, but is 10⁴ times less than the probability of death due to an accident). Given this stringent failure criterion, the desired value is only attainable by introducing safety factors with a pre-knowledge from the randomness of the material and its distribution. A probabilistic framework based on the size effect and uncertainty of existing concrete strength was proposed by Le and Bažant [35]. To validate this approach, four coefficients of variation (COV), i.e., 5%, 10%, 25%, and 40% were considered for a diagonal failure of RC beams in shear. It was highlighted that size effect significantly affects failure probability, thus if a large beam is designed based on the results of small beams, size effect consideration must be accounted for in safety factors. It is important to note that there are existing methods based on the deep neural network (DNN) that correlate experimental data to uncertainty analysis and their forward solution-solving approach does not require classical discretization as they directly solve with DNN [40–42].

A holistic review of the above-mentioned references and studies reveal that the majority used the lattice approach to simulate the behavior on the meso-scale and/or to account for material transport. To the best knowledge of the authors, limited studies regarding the application of the lattice model on the structural level [43–46] exist. Hence, this study aims adding to the existing literature regarding the structural application of the lattice model. Advantages of lattice modeling are as follows.

1) Creating random uncertainties and introducing deficiencies to integral points of a finite element mesh requires either advanced techniques or complex coding. Lattice modeling is a more appropriate choice for the objectives of this study since it makes it easier to introduce deficiencies to lattice elements than to nodes of finite elements in a continuum media.

2) Lattice modeling is less time-consuming than common finite element simulations.

2 Comparative study

2.1 Introduction

Before addressing the topic of this research, validation studies were carried out on flexure- and shear-deficient beams [47]. The beam with 50% deficiency in flexure (FD2-50%) was simulated and two 10-mm rebars were used in the tension region (compared to three 12-mm rebars in the original beam). Grade 500 steel rebars with a yield stress and ultimate stress of 500 and 545 MPa, respectively, were used as rebars. Similarly, the 60% shear-deficient beam (SD3-60%) was also modeled. Eight-millimeter stirrups with 750 mm spacing were used for this purpose (compared to 120 mm in the original beam). The term “deficiency” in comparative studies refers to inadequate longitudinal rebars in flexure and inadequate shear reinforcement in shear, which is different from the “deficiency” used in lattice modeling where it refers to the reduction in the compressive strength of the lattice element. The overall span of the beam was 1800 mm (clear span: 1500 mm) and its cross-section was 150200 mm. There were two 8-mm hanger rebars in the compression zone of the beams. The 28-d characteristic cubic compressive strength of concrete was 40 MPa, which was cast using ordinary Portland cement, natural sand, and coarse aggregates with a maximum size of 10 mm. Both beams were tested under four-point loading. Geometric details of the beams according to Banjara and Ramanjaneyulu [47] are given in Fig.1(a) and Fig.1(b). Once the adopted numerical procedure, given in the following sections, is validated, uncertainties in the number of deficient members and levels of deficiency will be discussed. It should be highlighted that the main purpose of this study is simulation using lattices and FEM software, which serve as an additional measure to verify the overall failure pattern of lattice modeling. Therefore, this study only provides a brief description of the finite element software since the focal point of the manuscript concerns lattice modeling.

2.2 Numerical simulation

2.2.1 ATENA

ATENA is an advanced finite element software specifically developed for the simulation of concrete structures and its capabilities have been confirmed by numerous studies. For example, this software has been used to carry out flexural [48–50], shear [51,52] and uncertainty studies [53,54].

2.2.1.1 Concrete material model

1) Nonlinear Cementitious2 and uniaxial concrete model

“Nonlinear Cementitious2” concrete model was used, which is a plastic-fracture based model that accounts for both plastic behavior and tensile cracking of concrete. The fracture model follows the classic smeared crack approach and the crack band model. It uses the Rankin’s failure criterion and exponential softening, which can either be a fixed or rotated crack model. The hardening and softening behavior of the plastic model uses the Menetrey−Willam [55] failure surface. The uniaxial behavior of concrete is shown in Fig.2(a).

2) Biaxial concrete model

“Nonlinear Cementitious2” concrete model uses the biaxial failure criterion of Kupfer et al. [57]. Fig.2(b) shows the biaxial behavior of concrete.

3) Triaxial concrete model

The behavior of concrete under triaxial stress follows the Menetrey−Willam [55] failure surface. Fig.2(c) shows the triaxial behavior of concrete. For further information regarding the constitutive models, symbols, etc., please refer to the ATENA documentation [56].

2.2.1.2 Mesh and material models

Eight-node 3D hexahedral elements account for the characteristic length of tension, compression, and strain localization. A bilinear strain-hardening stress-strain model was used to simulate longitudinal and transverse reinforcements embedded in concrete as 1D truss elements. A linear elastic model stress-strain curve was used to model steel supports and loading plates with thicknesses of 40 mm as eight-node 3D hexahedral elements. Input parameters were based on experimental data provided by Banjara and Ramanjaneyulu [47]. The beams were simply supported and an overall mesh size of 25 mm was used in three orthogonal directions. Loading was applied in a displacement-controlled manner with a rate of 0.1 mm per step of loading until failure occurred. Monitors were used to record reaction values and the mid-span deflection of the curve.

2.2.2 ABAQUS

A similar procedure to Section 2.2.1.1 was adopted to model the beam in ABAQUS. Definition of the materials and meshing followed the explanation given in Section 2.2.1.1. Tensile and compressive damage were also taken into account by adopting the concrete damage plasticity model [58] according to Fig.3. For further information and definition of symbols, please refer to Ref. [58]. The input parameters are given in Tab.1.

2.2.3 Lattice

Following the fundamental principles introduced by Hrennikoff [30] and based on the studies by Miki and Niwa [59], the beam tested by Banjara and Ramanjaneyulu [47] was discretized into 2D truss elements and the cross-sections of each horizontal, vertical, and diagonal members were calculated based on Eq. (1):

(1)\begin{array}{rl}& A_{\mathrm{v}}=\frac{3}{8}\frac{3k^2-1}{k}at;A_{\mathrm{h}}=\frac{3}{8}(3-k^2)at;\\ & A_{\mathrm{d}}=\frac{3}{16}\frac{{(1+k^2)}^{3/2}}{k}at,\end{array}

where t is the thickness of the plate; a refers to the vertical length of the lattice block, and at is their multiplication (at); other parameters are presented in Fig.4. Stress−strain curves were adopted for lattice simulation. Their corresponding equations for each curve segment and the methodology flowchart for lattice simulation in MATLAB software are given in Fig.5, Eqs. (2) and (3) and Fig.6, respectively. Firstly, mesh sensitivity analyses were performed by changing the mesh size from 100 to 5 mm. Results for mesh sizes 5, 10, 20, and 40 mm for the lattice model are given in Fig.7. It is worth noting that for the mesh size equal to 80 mm in shear, beams deflected to a maximum of 1 mm and numerical instabilities ensued immediately. Hence, they were not presented. Shear-dominant beams were sensitive to mesh size compared to their flexure-dominant counterparts. The observed results were similar for mesh sizes of 10 mm or less in lattice models, thus a 10 mm mesh size was adopted for horizontal, vertical, and oblique members.

The comparison of failure patterns in Fig.8(a)–Fig.8(d), load−deflection curves for flexure in Fig.9, and its corresponding results in Fig.10(a)–Fig.10(d) and Fig.11 for shear show that the results demonstrate a high level of agreement, validating the numerical procedure for the lattice simulation. It is also noteworthy that the computational effort is much lower in lattice modeling compared to that in ATENA and ABAQUS, as seen from Tab.2. In addition, the load capacities are very similar to the experimental results given in Tab.3.

\mathrm{O}\mathrm{H}:{\sigma }_i=f_{\mathrm{c}}^{\prime }\left[2\left(\frac{{\epsilon }_i}{{\epsilon }_0}\right)-{\left(\frac{{\epsilon }_i}{{\epsilon }_0}\right)}^2\right],\mathrm{A}\mathrm{B}:{\sigma }_i=E_{\mathrm{C}0}\left({\epsilon }_i-\frac{{\epsilon }_{\mathrm{B}}^2}{2{\epsilon }_0}\right),

\mathrm{C}\mathrm{E}:{\sigma }_i=E_{\mathrm{C}0}\left({\epsilon }_i-\frac{{\epsilon }_{\mathrm{C}}^2}{2{\epsilon }_0}\right),\mathrm{E}\mathrm{G}:{\sigma }_i=E_{\mathrm{C}0}\cdot {\epsilon }_{\mathrm{t}}{\left(\frac{{\epsilon }_{\mathrm{t}}}{{\epsilon }_i-{\epsilon }_{\mathrm{K}}}\right)}^{0.4},

\mathrm{G}\mathrm{C}:\{\begin{array}{l}{\sigma }_i={\displaystyle \frac{2f_{\mathrm{c}}^{\prime }}{{\epsilon }_0({\epsilon }_{\mathrm{G}}-{\epsilon }_{\mathrm{C}})}}\left[a_2\left({\epsilon }_i-{\epsilon }_{\mathrm{C}}\right)-a_1\left({\epsilon }_i-{\epsilon }_{\mathrm{C}}\right)\right],\\ a_1=\left[{\epsilon }_{\mathrm{C}}-{\displaystyle \frac{{\epsilon }_{\mathrm{C}}^2}{2{\epsilon }_0}}\right],\\ a_2=\left[{\epsilon }_{\mathrm{t}}{\left({\displaystyle \frac{{\epsilon }_{\mathrm{t}}}{{\epsilon }_{\mathrm{G}}-{\epsilon }_{\mathrm{D}}}}\right)}^{0.4}\right],\end{array}

\mathrm{F}\mathrm{C}:{\sigma }_i=\frac{f_{\mathrm{c}}^{\prime }\left[2\left({\displaystyle \frac{{\epsilon }_i}{{\epsilon }_0}}\right)-{\left({\displaystyle \frac{{\epsilon }_i}{{\epsilon }_0}}\right)}^2\right]}{{\epsilon }_{\mathrm{C}}-{\epsilon }_{\mathrm{F}}}\left({\epsilon }_i-{\epsilon }_{\mathrm{G}}\right),

(2)\mathrm{O}\mathrm{I}:{\sigma }_i=E_{\mathrm{C}0}\cdot {\epsilon }_i;\mathrm{I}\mathrm{J}:{\sigma }_i=E_{\mathrm{C}0}\cdot {\epsilon }_{\mathrm{t}}{\left(\frac{{\epsilon }_{\mathrm{t}}}{{\epsilon }_i}\right)}^{0.4},

(3)\begin{array}{l}\mathrm{O}\mathrm{A}:{\sigma }_i=E{\epsilon }_i,\mathrm{A}\mathrm{B}:{\sigma }_i={\displaystyle \frac{E{\epsilon }_i+(v-1)f_{\mathrm{y}}}{v}},\\ \mathrm{B}\mathrm{C}:{\sigma }_i=E{\epsilon }_i-{\displaystyle \frac{(v-1)}{v}}\left(E{\epsilon }_{\mathrm{B}}-f_{\mathrm{y}}\right),\\ \mathrm{C}\mathrm{D}:{\sigma }_i={\displaystyle \frac{E{\epsilon }_i-(v-1)f_{\mathrm{y}}}{v}},\\ \mathrm{D}\mathrm{E}:{\sigma }_i=E{\epsilon }_i-{\displaystyle \frac{(v-1)}{v}}\left(E{\epsilon }_{\mathrm{B}}+f_{\mathrm{y}}\right),\end{array}

where ε_i refers to strains; σ_i denotes stress; E denotes modulus of elasticity; f_{\mathrm{c}}^{\prime } is the compressive strength of concrete; f_y denotes yield stress of steel, and v denotes Poisson ratio.

3 Parametric studies

3.1 Flexure-dominant beams

3.1.1 Load−deflection curves

Following the validation studies presented in Section 2, uncertainty and the stochastic nature of the compressive strength deficiency of concrete were accounted for. Three levels of material deficiency, i.e., 10%, 20%, 30% were randomly introduced to 5%, 10%, 15%, and 20% of members. For brevity, specimens that failed in flexure were denoted with the letter “F” and those that failed in shear were denoted with the letter “S”. To distinguish between the different models, the letters “M” and “S” were used to symbolize the fraction of elements with deficiency and the level of strength deficiency, respectively. For example, F-M5S10 and S-M5S10 denote specimens with 5% of members and 10% deficient in compressive strength in flexure and shear, respectively. Due to symmetry, only half of the beam was modeled. As presented in Fig.12, a total of 1000 analyses were carried out for each beam series in flexure and the maximum load value was recorded (the y-axis of each plot is the relative average load with respect to the beam with no deficiencies). The use of 1000 analyses was chosen from a statistical perspective to provide a large database, making the results more reliable. It can be seen from the database of Fig.12 and the average load−deflection curves in Fig.13(a) (due to similarity, only the results of one model was presented) that the number of the elements with compressive strength deficiency has a greater impact on load capacity compared to the level of compressive strength deficiency in those elements. This observation is further corroborated by the statistics illustrated in Fig.13(b) and Tab.4. The correlation map of the deficiencies against the relative average load reveals that the correlation between member deficiency and the relative average load falls within the range of [–1,–0.818] and [–0.445,–0.273] for strength deficiency. According to these values, where +1 represents the best positive and –1 represents the worst negative correlation, the higher negative impact of member deficiency is inferred. It is observed that, relative to the specimen without deficiency (M0S0), a minimum of 1% and a maximum of 32% decrease occurred in specimens F-M5S10 and F-M20S30, respectively.

3.1.2 Statistical distributions and cracking pattern

According to Tab.5 and Tab.6, as well as the statistical distributions given in Fig.14, irrespective of the level of member and strength deficiencies, distributions followed a normal or pseudo-normal distribution with an increasing standard deviation as the member and strength deficiencies increased. Fig.15 shows a typical cracking pattern of beams in flexure. As expected, cracks form within the pure-bending region and propagate toward the upper parts of the beam as load increases.

3.1.3 Energy absorption

Fig.16(a) and Fig.16(b) show that, irrespective of the level of member or strength deficiency, a linear relationship exists between the absorbed energy and mid-span deflection (for brevity, only the results of two models were presented).

3.1.4 Nonlinear regression

According to Ref. [60], a normalized flexural load-deflection curve should satisfy the following: 1) when x = 0, y = 0; 2) for x between 0 and 1, the slope of the ascending part is negative; 3) at peak load, x and y are equal to 1 and the slope of the curve is 0; 4) when x > 1 and {\displaystyle \frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}}=0, it is an inflection point at the descending part; 5) when x > 1 and {\displaystyle \frac{{\mathrm{d}}^{3}y}{\mathrm{d}{x}^3}}=0, it pinpoints the location of the maximum curvature in the descending branch; 6) as x approaches ∞, y approaches ∞ as well and the slope of the curve approaches zero; 7) when x ≥ 1, 0 ≤ y ≤ 1.

The model introduced by Wang and Xu [61] was evaluated by Wee et al. [62], which yielded good results (Eq. (4)):

(4)y=\frac{ax+b{x}^2}{1+cx+d{x}^2},

where a, b, c and d are unknown parameters obtained from the regression analysis. It worth noting that the model introduced by Wang and Xu [61] is physically meaningless as it is directly fitted to the experimental curve. Herein, Eq. (5) is proposed based on the load-deflection observations:

(5)y=\frac{a+bx}{1+cx+d{x}^2},

where x is the ratio of a given deflection to its corresponding value at the peak load; y is the ratio of a given load value to the peak load. Equation (5) should satisfy the following:

For any value of x or fitting parameters, y ≥ 0. Otherwise, assume y = 0.

It can be observed in Fig.17(a) and Fig.17(b), and Tab.7 that fitting curves demonstrate a high level of agreement with the numerical results with coefficient of determination ratios (R²) over 0.90 (for brevity, only the curves of two models were presented).

Similar to the nonlinear regression analyses carried out for the load−deflection curves, a 3D surface was fitted to the numerical results, as shown in Fig.18, which correlates the deficiencies in the number of members and level of strength to the normalized relative average load (relative to the specimen without deficiency). Equation (6) shows the formula of the 3D surface, which correlates well with the data.

(6)\begin{array}{rl}Z=& -\frac{8289}{{10}^7}{M}^2-\frac{3675}{{10}^8}{S}^2-\frac{2065}{{10}^7}MS+\frac{5971}{{10}^6}M\\ & -\frac{1509}{{10}^6}S-\frac{9864}{{10}^4},R^2=0.9691,\end{array}

where refers to the relative average load with respect to the specimen without deficiency, i.e., F-M0S0.

3.2 Shear-dominant beams

3.2.1 Load−deflection curves

Similar to the procedure adopted in Section 3.1, 1000 analyses were carried out for the shear-dominant beams (Fig.19). By drawing an analogy between the counterpart values for flexure, it is observed that the rate of reduction in flexure is greater than that of shear. For instance, for specimen F-M20S30 and S-M20S30, the load-bearing capacities were 32% and 21.5% lower with respect to the specimen without deficiency, respectively. It is also observed that the scatter of data is higher in flexure (range: 0.680–0.990, upper and lower values for F-M5S10 and F-M20S30, respectively) than that of shear (range: 0.795–0.996, upper and lower limits for S-M5S10 and M20S30, respectively). This observation can be justified by the fact that the bending moment covers a larger area along the length of the beam than shear force, hence the scatter of data and probability of obtaining lower load values is higher (Fig.20; for brevity, only the results of one model was presented).

Furthermore, the correlation of deficiencies with the mean load capacity in shear is similar to Fig.13(b) and falls within the range of [–1,–0.818]. For strength deficiency, it falls within the range of [–0.445,–0.273], but with slightly higher dependency on the number of deficient members (–0.966 vs. –0.947), as given in Tab.8. It is also observed that, relative to the specimen without deficiency, M0S0, a minimum of 1% and a maximum of 32% decrease occurred in specimens F-M5S10 and F-M20S30, respectively.

3.2.2 Statistical distributions and cracking pattern

As previously mentioned, the scatter of data in shear is less than that of flexure and the distribution of strengths follows a normal and/or pseudo-normal distribution, as seen from Fig.21 and Tab.9 and Tab.10.

Regarding the cracking patterns, cracks were formed between the supports and loading plate with an approximate 45° angle with respect to the horizontal direction (Fig.22).

3.2.3 Energy absorption

The linear relationship between the absorbed energy and deflection values observed in flexure also applies to shear, as seen from Fig.23(a) and Fig.23(b) (for brevity, only the curves of two models were presented).

3.2.4 Nonlinear regression

The same procedure detailed in Section 3.1.4 was adopted and values with a high degree of concordance were obtained using the same equation. The results are presented in Fig.24(a) and Fig.24(b), and Tab.11 (for brevity, only the curves of two models were presented).

Accordingly, a 3D curve was fitted to the obtained results to correlate the deficiency parameters to the relative average load value (Fig.25). The formula of the fitted surface is given in Eq. (7), which yields desirable values:

(7)\begin{array}{rl}Z=& \frac{1.237}{{10}^4}{M}^2-\frac{8.587}{{10}^6}{S}^2-\frac{1.098}{{10}^5}MS\\ & -\frac{1.557}{{10}^2}M-\frac{9.86}{{10}^4}S+1.093,R^2=\mathrm{0.9578.}\end{array}

4 Conclusions

A numerical procedure based on the theories proposed by Hrennikoff [30] was used to simulate RC beams failing in flexure and shear using 2D lattice models. The procedure was initially validated against available literature and two FEM software (which served as additional validation). Subsequently, to account for the uncertainty and randomness of the compressive strength of concrete, different levels of deficiencies both in the number of deficient members and level of strength deficiencies were accounted for by carrying out 1000 analyses for each flexure- and shear-dominant beams (24000 analyses in total). The conclusions of this study are as follows.

1) Given the complexity of introducing deficiencies to the nodes or elements of a finite element model, lattice modeling is a powerful statistical tool to account for uncertainties. It is also less time-consuming and yields results similar to the experimental data, which were both presented for comparative purposes.

2) According to correlation studies, the number of compressive strength-deficient members has a greater impact on the load-bearing capacity of beams compared to the level of compressive strength deficiency.

3) Scatter of random load-capacities is higher in flexure (range: 0.680–0.990, upper and lower values for F-M5S10 and F-M20S30, respectively) than that of shear (range: 0.795–0.996, upper and lower limits for S-M5S10 and M20S30, respectively) as bending moments are present throughout the beam, while shear forces only exist in one-third of both ends. Hence, the possibility of deficient members is higher in flexure.

4) Regardless of the level of deficiency in either the number of members or strength, energy absorption values vary linearly with the mid-span deflection.

5) For the beams studied herein, the equations proposed for nonlinear regression models successfully captured the load−deflection response of beams, while accounting for the simultaneous contribution of deficiency parameters with values over 0.90.

It is also important to emphasize the scarcity of research regarding the use of lattice modeling on the structural level and given its advantageous nature in carrying out statistical work and accounting for uncertainty, we would like to push for lattice modeling applications in the simulation of other structural members.

References

[1] Hamad W I, Owen J S, Hussein M F. An efficient approach of modelling the flexural cracking behavior of un-notched plain concrete prisms subject to monotonic and cyclic loading. Engineering Structures, 2013, 51: 36–50

[2] Bin S, Li Z. Multi-scale modeling and trans-level simulation from material meso-damage to structural failure of reinforced concrete frame structures under seismic loading. Journal of Computational Science, 2016, 12: 38–50

[3] Yılmaz O, Molinari J F. A mesoscale fracture model for concrete. Cement and Concrete Research, 2017, 97: 84–94

[4] Mauludin L M, Oucif C. Computational modeling of fracture in concrete: A review. Frontiers of Structural and Civil Engineering, 2020, 14(3): 586–598

[5] Birck G, Iturrioz I, Riera J D, Miguel L F. Influence of mesh orientation in discrete element method simulations of fracture processes. Journal of Strain Analysis for Engineering Design, 2018, 53(6): 400–407

[6] Shishegaran A, Khalili M R, Karami B, Rabczuk T, Shishegaran A. Computational predictions for estimating the maximum deflection of reinforced concrete panels subjected to the blast load. International Journal of Impact Engineering, 2020, 139: 103527

[7] Shishegaran A, Karami B, Rabczuk T, Shishegaran A, Naghsh M A, Mohammad Khani M. Performance of fixed beam without interacting bars. Frontiers of Structural and Civil Engineering, 2020, 14(5): 1180–1195

[8] Shishegaran A, Varaee H, Rabczuk T, Shishegaran G. High correlated variables creator machine: Prediction of the compressive strength of concrete. Computers & Structures, 2021, 247: 106479

[9] Naghsh M A, Shishegaran A, Karami B, Rabczuk T, Shishegaran A, Taghavizadeh H, Moradi M. An innovative model for predicting the displacement and rotation of column-tree moment connection under fire. Frontiers of Structural and Civil Engineering, 2021, 15(1): 194–212

[10] Shishegaran A, Ghasemi M R, Varaee H. Performance of a novel bent-up bars system not interacting with concrete. Frontiers of Structural and Civil Engineering, 2019, 13(6): 1301–1315

[11] Shishegaran A, Moradi M, Naghsh M A, Karami B, Shishegaran A. Prediction of the load-carrying capacity of reinforced concrete connections under post-earthquake fire. Journal of Zhejiang University. Science A, 2021, 22(6): 441–466

[12] Shishegaran A, Karami B, Danalou E S, Varaee H, Rabczuk T. Computational predictions for predicting the performance of steel 1 panel shear wall under explosive loads. Engineering Computations, 2021, 38(9): 3564–3589

[13] Karami B, Shishegaran A, Taghavizade H, Rabczuk T. Presenting innovative ensemble model for prediction of the load carrying capacity of composite castellated steel beam under fire. Structures, 2021, 33: 4031–4052

[14] Pandey V B, Singh I V, Mishra B K, Ahmad S, Rao A V, Kumar V. Creep crack simulations using continuum damage mechanics and extended finite element method. International Journal of Damage Mechanics, 2019, 28(1): 3–4

[15] Rabczuk T, Belytschko T. Cracking particles: A simplified meshfree method for arbitrary evolving cracks. International Journal for Numerical Methods in Engineering, 2004, 61(13): 2316–2343

[16] Rabczuk T, Belytschko T. A three-dimensional large deformation meshfree method for arbitrary evolving cracks. Computer Methods in Applied Mechanics and Engineering, 2007, 196(29-30): 2777–2799

[17] Hohe J, Gall M, Fliegener S, Hamid Z M. A continuum damage mechanics model for fatigue and degradation of fiber reinforced materials. Journal of Composite Materials, 2020, 54(21): 2837–2852

[18] Wang J, Guo X, Zhang N. Study of the progressive failure of concrete by phase field modeling and experiments. International Journal of Damage Mechanics, 2021, 30(9): 1377–1399

[19] Ren H L, Zhuang X Y, Anitescu C, Rabczuk T. An explicit phase field method for brittle dynamic fracture. Computers & Structures, 2019, 217: 45–56

[20] Hentz S, Daudeville L, Donzé F V. Identification and validation of a discrete element model for concrete. Journal of Engineering Mechanics, 2004, 130(6): 709–719

[21] RabczukTZiGBordasSNguyen-XuanH. A simple and robust three-dimensional cracking-particle method without enrichment. Computer Methods in Applied Mechanics and Engineering, 2010, 199(37−40): 2437−2455

[22] Oliver-Leblond C. Discontinuous crack growth and toughening mechanisms in concrete: A numerical study based on the beam-particle approach. Engineering Fracture Mechanics, 2019, 207: 1–22

[23] CundallP A. A computer model for simulating progressive, large-scale movement in blocky rock system. In: Proceedings of the International Symposium on Rock Mechanics. Madrid: International Society for Rock Mechanics, 1971

[24] Potyondy D O, Cundall P A. A bonded-particle model for rock. International Journal of Rock Mechanics and Mining Sciences, 2004, 41(8): 1329–1364

[25] GoodmanR E. Discontinuous deformation analysis—A new method for computing stress, strain and sliding of block systems. In: 29th US Symposium on Rock Mechanics (USRMS). Minneapolis, MN: OnePetro, 1988

[26] Cherkaev A, Ryvkin M. Damage propagation in 2D beam lattices: 1. Uncertainty and assumptions. Archive of Applied Mechanics, 2019, 89(3): 485–501

[27] Bolander J E Jr, Saito S. Fracture analyses using spring networks with random geometry. Engineering Fracture Mechanics, 1998, 61(5−6): 569–591

[28] D’Addetta G A, Kun F, Ramm E. On the application of a discrete model to the fracture process of cohesive granular materials. Granular Matter, 2002, 4(2): 77–90

[29] Nikolić M, Karavelić E, Ibrahimbegovic A, Miščević P. Lattice element models and their peculiarities. Archives of Computational Methods in Engineering, 2018, 25(3): 753–784

[30] Hrennikoff A. Solution of problems of elasticity by the framework method. Journal of Applied Mechanics, 1941, 8: 169–175

[31] Zhao G F, Fang J, Zhao J. A MLS-based lattice spring model for simulating elasticity of materials. International Journal of Computational Methods, 2012, 9(3): 1250037

[32] Jiang C, Zhao G F, Khalili N. On crack propagation in brittle material using the distinct lattice spring model. International Journal of Solids and Structures, 2017, 118: 41–57

[33] Aydin B B, Tuncay K, Binici B. Simulation of reinforced concrete member response using lattice model. Journal of Structural Engineering, 2019, 145(9): 04019091

[34] Aydin B B, Binici B, Tuncay K. Lattice simulation of concrete compressive behavior as indirect tension failure. Magazine of Concrete Research, 2021, 73(8): 394–409

[35] Le J L, Bažant Z P. Failure probability of concrete specimens of uncertain mean strength in large database. Journal of Engineering Mechanics, 2020, 146(6): 04020039

[36] Yeh I C. Modeling of strength of high-performance concrete using artificial neural networks. Cement and Concrete Research, 1998, 28(12): 1797–1808

[37] Reineck K H, Bentz E C, Fitik B, Kuchma D A, Bayrak O. ACI-DAfStb database of shear tests on slender reinforced concrete beams without stirrups. ACI Structural Journal, 2013, 110(5): 867–875

[38] Dönmez A, Bažant Z P. Size effect on punching strength of reinforced concrete slabs with and without shear reinforcement. ACI Structural Journal, 2017, 114(4): 875–886

[39] MelchersRE. Structural Reliability: Analysis and Prediction. New York: John Wiley and Sons, 2018

[40] Nguyen-Thanh V M, Anitescu C, Alajlan N, Rabczuk T, Zhuang X. Parametric deep energy approach for elasticity accounting for strain gradient effects. Computer Methods in Applied Mechanics and Engineering, 2021, 386: 114096

[41] Samaniego E, Anitescu C, Goswami S, Nguyen-Thanh V M, Guo H, Hamdia K, Zhuang X, Rabczuk T. An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications. Computer Methods in Applied Mechanics and Engineering, 2020, 362: 112790

[42] Goswami S, Anitescu C, Chakraborty S, Rabczuk T. Transfer learning enhanced physics informed neural network for phase-field modeling of fracture. Theoretical and Applied Fracture Mechanics, 2020, 106: 102447

[43] Miki T, Niwa J. Three-dimensional lattice model analysis of RC columns subjected to seismic loads. Special Publication, 2006, 237: 241–258

[44] Simão M R, Miki T. Concrete Repair, Rehabilitation and Retrofitting IV. Boca Raton, FL: CRC Press, 2015, 214–215

[45] SIMÃO M R, Tomohiro M I. Dynamic analysis for RC columns with circular cross section using multi-directional polygonal 3D lattice model. Proceeding of JCI, 2015, 2: 757–762

[46] Kwon M, Jeong Y, Kim J. Numerical study of RC beam-column joints using a 4-node lattice element analysis method. KSCE Journal of Civil Engineering, 2021, 25(3): 960–972

[47] Banjara N K, Ramanjaneyulu K. Effect of deficiencies on fatigue life of reinforced concrete beams. ACI Structural Journal, 2020, 117(3): 31–44

[48] Sadaghian H, Pourbaba M, Andabili S Z, Mirmiran A. Experimental and numerical study of flexural properties in UHPFRC beams with and without an initial notch. Construction & Building Materials, 2021, 268: 121196

[49] Nariman N A, Hamdia K, Ramadan A M, Sadaghian H. Optimum design of flexural strength and stiffness for reinforced concrete beams using machine learning. Applied Sciences (Basel, Switzerland), 2021, 11(18): 8762

[50] Kian N, Farzam M, Rezaie Oshtolagh M. Carbon fiber reinforced polymer strengthened reinforced concrete square columns under pre-existing eccentric loads. Advances in Structural Engineering, 2021, 24(15): 3420–3432

[51] Farzam M, Sadaghian H. Mechanical model for punching shear capacity of rectangular slab-column connections. Structural Concrete, 2018, 19(6): 1983–1991

[52] Farzam M, Sadaghian H, Khodadade G. Shear behaviour of elongated rectangular wall–footing connections under eccentric loads. Magazine of Concrete Research, 2019, 71(1): 43–54

[53] Rimkus A, Cervenka V, Gribniak V, Cervenka J. Uncertainty of the smeared crack model applied to RC beams. Engineering Fracture Mechanics, 2020, 233: 107088

[54] Zięba J, Buda-Ożóg L, Skrzypczak I. Probabilistic method and FEM analysis in the design and analysis of cracks widths. Engineering Structures, 2020, 209: 110022

[55] Menetrey P, Willam K J. Triaxial failure criterion for concrete and its generalization. Structural Journal, 1995, 92(3): 311–318

[56] ČervenkaVJendeleLČervenkaJ. ATENA Program Documentation Part 1—Theory. Prague: Červenka Consulting Engineers, 2021

[57] Kupfer H, Hilsdorf H K, Rusch H. Behavior of concrete under biaxial stresses. Journal proceedings, 1969, 66(8): 656–666

[58] ABAQUSSoftware. ABAQUS User Analysis Manual. Providence, RI: Dassault Systems Simulia Corp., 2016

[59] Miki T, Niwa J. Nonlinear analysis of RC structural members using 3D lattice model. Journal of Advanced Concrete Technology, 2004, 2(3): 343–358

[60] GuoZ. Strength and Deformation of Concrete: Test Basis and Constitutive Relation. Beijing: Tsinghua University Press, 1997

[61] Wang L M, Xu S L. Characteristic curve of concrete and fiber reinforced concrete. Journal of Dalian University of Technology, 2002, 42(5): 580–585

[62] Wee T H, Chin M S, Mansur M A. Stress-strain relationship of high-strength concrete in compression. Journal of Materials in Civil Engineering, 1996, 8(2): 70–76

RIGHTS & PERMISSIONS

Higher Education Press