Knowledge-Based Prediction of Load-Carrying Capacity of RC Flat Slab through Neural Network and FEM

Ahmad, Afaq; Arshid, Muhammad Usman; Mahmood, Toqeer; Ahmad, Naveed; Waheed, Abdul; Safdar, Syed Shujaa

doi:https://doi.org/10.1155/2021/4528945

Mathematical Problems in Engineering

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Artificial Intelligence for Civil Engineering 2021

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Research Article | Open Access

Volume 2021 | Article ID 4528945 | https://doi.org/10.1155/2021/4528945

Knowledge-Based Prediction of Load-Carrying Capacity of RC Flat Slab through Neural Network and FEM

Afaq Ahmad,¹Muhammad Usman Arshid,¹Toqeer Mahmood,²Naveed Ahmad,¹Abdul Waheed,³and Syed Shujaa Safdar⁴

Academic Editor: Danial Armaghani

Received23 Jun 2021

Revised01 Sept 2021

Accepted14 Sept 2021

Published29 Sept 2021

Abstract

The present research work aims to compare the results for predicting the ultimate response of Reinforced Concrete (RC) members using Current Design Codes (CDCs), an alternative method based on the Compressive Force Path (CFP) method, and Artificial Neural Network (ANN). For this purpose, the database of 145 samples of RC Flat Slab with the simple supported condition under concentrated load is developed from the latest published work. All the cases studied were Square Concrete Slabs (SCS). The critical parameters used as input for the study were column dimension, c_s, depth of the slab, d_s, shear span ratio, , longitudinal percentage steel ratio, ρ_ls, yield strength of longitudinal steel, f_yls, the compressive strength of concrete, f_cs, and ultimate load-carrying capacity, V_us. Seven ANN models were trained using different combinations of input parameters and different points of hidden neurons with different activation functions. The results exhibited that SCS-4 was the most optimized ANN model, having the maximum value of R (89%) with the least values of MSE (0.62%) and MAE (6.2%). It did not only reduce the error but also predicted accurate results with the least quantity of input parameters. The predictions obtained from the studied models (i.e., CDCs, CFP, and ANN) exhibited that results obtained using the ANNs model correlated well with the experimental data. Furthermore, the FEM results for the selected cases show the closer result to the ANN predictions.

1. Introduction

The practice of Reinforced Concrete (RC) flat slabs or flat plate or waffle slabs in which the RC slab is placed directly over the RC members (as explained in Figure 1) is very common in the construction industry. It improves the structural flexibility and reduces the overall height of the structure, facilitating the provision of an additional number of floors (as illustrated in Figure 2). It also makes construction easier by reducing the shuttering time and relevant costs. However, due to the direct placement of RC slab over columns and because of its simple shape, the flat slabs are more vulnerable to a sudden or punching failure [1]. Hentschel [1] provided a detailed explanation of the work done in the year 2001–2005 about the problem of punching failure of RC flat under monotonic uploads. The other researchers also worked on the validity of these methods by comparing their results with the experimental values and developed a database with the important information of all critical parameters that accounted for the punching failure [2, 3].

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The actual construction of slab-column connection is an important problem in the construction of RC flat slab structures due to difficulties related to operating services and higher concentration of structural load in the slab-column region (as shown in Figure 3(a)). Such load concentration in the vicinity of slab-column joint may result in shear failure especially on the 2D members [2, 3]. This type of concrete-based failure causes a rapid failure without noteworthy warning (as illustrated in Figure 3(b)), indicating the succeeding collapse of the part of the car parking building at Wolverhampton [4, 5].

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In current design practice, there are certain doubts regarding the use of flat slabs in earthquake-resistant RC structures. The requirements of Eurocode 8 [2, 3] do not include the case of flat slab or flat plate supported on RC members as part of the resisting system against the lateral loading. On the contrary, ACI 318 [2, 3] includes guidelines for the analysis and assessment of RC flat slabs as part of an “intermediate moment frame” but excludes them from use in “special moment frames” [2, 3]. The results of insufficient predicting behavior by current design codes were experienced in different earthquake events [2, 3]. In an attempt to satisfy the above need, the alternative new design methodology known as the Compressive Force Path (CFP) methodology [6] is used in this comparative study. In comparison with the CDCs, the CFP methodology is based on the use of simple failure criteria. Although the latter criteria are developed from primary assumptions while excluding the requirement of the test data, their efficiency for evaluating shear capacity based on lab data is less than that of CDCs but they provide a reasonable indication for the causes and location of punching initiation. Moreover, the experimental verification of the method has shown that, in comparison with CDCs, it is capable of providing design results that fully satisfy the routine requirements of CDCs for the design of earthquake-resistant RC structures [2, 3].

Artificial Neural Network (ANN) is inconsistent and inclined to develop strange connections among system variables, even if they are unaware of the type of interaction [7–11], which do not depend on the underlying mechanism assumptions [12–15]. In the last two decades, the use of Soft Computing (SC) methods to solve complex problems has been proposed by many researchers [16–35]. In this work, response of both complex and simple reinforced concrete members at ULR has been assessed by using CDC, CFP, and ANN techniques. In previous studies, ANNs have been used to estimate material behavior [36–39] as well as response of RC members [11, 40–48]. ANNs have achieved the advanced attention of the researcher for solving the problems, especially for estimating the ULR of composite concrete members (CCM) [49–51]. Also, load-carrying capacity of advance materials has been predicted using ANNs as discussed in the articles [52–75]. The other technique to determine the ultimate response of the RC members with FRP rebars is finite element analysis (FEA). The literature showed that many researchers worked on the FEM studies [76–80]. Also, load-carrying capacity of some advance materials has been predicted using FEM [81, 82].

The present work aims to compare the results obtained by using Compressive Force Path method (CFP), Current Design Codes (CDCs), and Artificial Neural Network (ANN) for estimating the response of the reinforced concrete (RC) Members at ULR. The comparative study exhibited that these codes predicted the underestimated values [11, 40–48]. Therefore, in this study, ANN models have been developed for predicting real load-carrying capacity of reinforced concrete flat slab. For this purpose, data for 145 samples of Square Concrete Slabs (SCSs) tested under concentric loading was collected from previous studies. The data included all the detailed information of column dimension, c_s, depth of the slab, d_s, shear span ratio, , longitudinal percentage steel ratio, ρ_ls, yield strength of longitudinal steel, f_yls, compressive strength of concrete, f_cs, and ultimate load-carrying capacity, V_us. Seven ANN models were developed and an optimized model was selected based on maximum R value, higher prediction level, and minimum MSE, MAE, and error values. (i.e., SCS-4 c_s/d_s,, ρ_ls, and f_cs/f_ys). Also, the predicted best model (i.e., SCS-4) exhibited reasonable accurate results to the lab values as compared to other counterparts i.e., CDCs. Furthermore, finite element modeling (FEM) is also carried out for the validation of the ANN prediction, for the selected cases. The FEA results are within reasonable limits with the experimental values and ANN results.

2. Physical Models for RC Flat Slab

The theory providing the base for load transfer at the ultimate limit response (ULR) of SCS, as available in current design handbooks, is constructed on the mechanics “Truss Analogy” models [40]. The equations of these design codes EC2 and ACI [2, 3] are empirical in nature, which varies to the data fitting and may result in failure such as the collapse of structures [5] (as shown in Figures 3(a) and 3(b)). However, the calculations of the CDCs appear to be significantly different from their counterpart experimental values. The major reason for this is the different nature of the analysis formulas used in the CDCs. However, these CDCs are used by professionals in engineering to test the capability of the flat slab-column connection at the ULR due to the ease of construction.

The CDCs assumptions [2, 3] have been based so far on the discussion of various models and their application to predict the response of the SCS at ULR. These models are based on a few important parameters i.e., column dimension, c_s, depth of the slab, d_s, shear span ratio, , longitudinal percentage steel ratio, ρ_ls, yield strength of longitudinal steel, f_yls, compressive strength of concrete, f_cs, and ultimate load-carrying capacity, V_us, accounted for the punching failure of SCS. The equation used in the ACI 318 [3] for predicting the SCS response as shown in Figure 4(a) iswhere

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And, the equation used in the EC2 [2] for predicting the SCS response as shown in Figure 4(b) iswherewhere is the steel ratio along the x-direction of slab and is the steel ratio along the y-direction of slab.

Beside the Truss Analogy (TA) methodology for the CDCs, the other counterpart theory is the CFP method [6] based on the an arch-like frame structure response at ULR [6] and illustrated in Figure 5, which is described by following equations:

3. Artificial Neural Network (ANN)

Neural networks simulate the human and animal nervous systems functioning on important information in the brain [8, 83, 84]. These networks are performed to assess the purposes based on extensive given information. ANNs can acquire, classify, summarize, and guess the assigned task because they can keep the information presented to them during the training process into their memory and also because of their adaptability. They are composed of numerous connected layers, each of which contains the interconnected neuron system. There is a tie (having numerical value) between every two neurons in continuous layers, as shown in Figure 6. The prediction made by the neuron is multiplied by these weights. In this latter process, the prediction of the neuron is passed through the link and is added to the bias, as shown in Figure 6. An ANN created on the experience of the Multilayer Feedforward ANN (MLFNN) is considered to be the most suitable for these types of problems. MLFNN has an input (information) layer, an output (target) layer, and one or more hidden layer(s) [85, 86]. Interestingly, these neurons are not interlinked within the same layer but are linked with the other neurons of the other layers [85, 86]. During many training cycles to reduce error values, different techniques were used, changing the weight of each link [9–13]. To reduce the error, the cross-validation technique is used by dividing the input data into three subsets data as discussed in the previous work of the author [9–13]. In this case, it can be said that the network has learned the function of a particular goal. As the name of the algorithm suggests, errors are propagated back from the output node to the input node. ANN’s chosen architecture describes the quantity of hidden layer(s) and the quantity of neuron(s) in each layer(s), as illustrated in Figure 6. Equation (7) describes the mathematical function of the artificial neuron aswhere O value is predicted using ANN, x_z represents assigned input values, corresponds to link values, and extraparametric values are represented by b.

As shown in Figure 6, the results of the activation are then transferred to the next layer. Then, based on available information, final selected weights are calculated. In the present and past studies, the authors used ANNs [42, 45, 48, 87, 88] to estimate the load-bearing capacity of the materials and individual RC members [42, 45, 48, 87, 88]. In the current study, tanh and hyperbolic activation functions are used between input and intermediate layers, while hyperbolic functions have been used between intermediate and output layers. The inaccuracy occurring during the process can be counted using equation (8). Once ANN is trained, equation (8) is used to compare the results obtained with already known target values:where T and O are target predicted values, respectively.

3.1. Database of SCS for ANN Modelling

To predict punching strength of RC flat slabs (SCS), the database was comprised of 145 SCS samples, as shown in Figure 7. The data regarding critical parameters, i.e., column dimension, c_s, depth of the slab, d_s, shear span ratio, , longitudinal percentage steel ratio, ρ_ls, longitudinal steel yield strength, f_yls, concrete compressive strength, f_cs,, and ultimate load-carrying capacity, V_us, is provided in Table 1. Out of 145 samples, 99% were failed in the shear manner and remaining in flexure. The histogram for critical parameters has been described in Figure 8(a). The figure described different parametric frequencies for certain sample values. It also defined limitations for ANN models that were to predict against limits of different parameters. The sensitivity analysis is carried out by using the correlation of the critical parameters as exhibited in Figure 8(b).

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3.2. Preprocessing Phase

ANN training gets significantly affected by standard data points because regardless of the unit system followed, all the different inputs at this point become common among the same values [44, 45]. Therefore, by using equation (9), all the critical parameters used in this work are eventually normalized between 0.1 and 0.9:where x is the current value, X is the normalized/uniform value, and Δ is the difference of limits.

In the present case, the value of the X_max = 0.9 and ΔX = 0.8 are used against the new limits of 0.1 to 0.9. Pearson’s correlation coefficient (R), as given by equation (10), was used to make different combinations of parameters [42, 45, 48, 87, 88]. Importance of the output parameter is determined based on difference of |R| values for input and output parameters:

For the modeling and training of ANN models, the author used the same procedures and guidelines as suggested in his own work [42, 45, 48, 87, 88]. For this purpose, a total of 7 different ANN models (as described in Table 2) with different combinations of the designed parameters for the SCS were used. The choice of these critical input parameters was built on the physical models i.e., CDCs which describe the machines under the SCS at ULR.

3.3. Optimized ANN Model

A multilayer ANN model was used to carry out the main features for the learning process of ANN. The MLFNN method of Levenberg–Marquardt was used by the authors for coding of the ANN model in the MATLAB environment [46]. The main ANN architecture as used by the authors was obtained from previous work of authors [11, 43]. The optimized ANN model was selected based on (i) error calculation using Pearson’s correlation for coefficient (R), (ii) mean absolute error (MAE), and (iii) mean squared error (MSE) [47, 48] which are analytically expressed by equations (10), (12) and (13), respectively:where Oi are the given target results, Ti are predicted results from ANN, and the number of all samples in the databank is represented by n [9–13].

Optimized ANN model is the one with lowest MSE and MAE values and highest R values [9–13].

Based on Figures 9 and 10, the SCS = 4 (ρ_ls, f_cs/f_yls,c_s/d_s, and ) is the best model fulfilling abovementioned criteria. However, like every other ANN model, this model was also used between the statistical information as discussed in Table 1. As the future work, more samples will be included from the available database to cover all the realistic sizes and material strength of the flat slab.

4. Comparative Studies of Flat Slab

In this section, the ANN-based predicted models, as discussed above, have been compared with the CFP method [52, 53] and with CDCs [1, 2], and the resulting data have been presented in Figures 11–13. Figure 11 presented the prediction results as the ANN, CFP, and CDCs against the experimental values (EXP). The results exhibited that, as compared with CDCs, the ANN and CFP are closer to the EXP results.

Figure 12 illustrates the normal distribution (ND) of the V_EXP/V_PRED ratios, for ANN (calculated as V_EXP/V_ANN) presenting the least standard deviation of 1.05 and was followed by CFP. After that, GD curves for CDCs (i.e., EC2 and ACI) have a wide range from 0 to 2 on the x-axis and a peak value smaller than 1.0 on the y-axis.

Overestimated and underestimated values for ULR have been presented in Figure 13. As observed before, the results obtained by using CDCs were lower than the provided ones, and 90% results against nonconventional models were ranged into the accurate region i.e.; between 0.755 and 1.255. The values obtained by other physical models were mostly underestimated compared to the ones obtained using ANN and CFP methods.

5. Parametric Studies of Flat Slab

This part is very important as it highlights the effect of the parametric studies against the prediction of CDCs (ACI and EC2) [2, 3], the CFP method [6], and the selected ANN model. The parametric investigation is done against the ratio of V_EXP/V_PRED for necessary design parameters only, i.e., parameters considered are column dimension, c_s, depth of the slab, d_s, shear span ratio, a_vs/d_s, longitudinal percentage steel ratio, ρ_ls, longitudinal steel yield strength, f_yls, concrete compressive strength, f_cs, and ultimate load-carrying capacity, V_us.

5.1. Effective Depth (d_s) and Shear-to-Depth against Ratios (a_vs/d_s)

The two most important geometric parameters for predicting the response of punching failure of the SCS are depth (d_s) and shear-to-depth against ratios (a_vs/d_s), as illustrated in Figures 14 and 15, respectively, for the case of the SCS. It is worth noting that, during the comparative study of the effect of (d_s) and (a_vs/d_s) against the suggested ratio of V_EXP/V_PRED, the results of ANN (V_ANN) are close to the experimental ratio, as illustrated in Figures 14 and 15. The scatter of the point for the other ratio (CDCs and CFP) V_EXP/V_ANN are far away from the line of unity. Furthermore, for the samples having a value between 75 mm < d < 135 mm and for the 5 </d < 10 showed the extensive deviation from the line of unity for the ratio V_EXP/V_PRED, as illustrated in Figures 14 and 15. Therefore, there is an urgent need to revise the CDC especially for the mentioned limits of the depth (d) and shear-to-depth against ratios (av/d).

5.2. Compressive Strength (f_c) and Longitudinal Steel Ratio ρ_l (%)

The two most important material parameters for predicting the response of punching failure of SCS are compressive strength (f_c) and longitudinal steel ratio ρ_l (%), as illustrated in Figures 16 and 17, respectively for the case of the SCS. It is worth noting to note that, during the comparative study of the effect of (f_c) and ρ_l (%) against the suggested ratio of V_EXP/V_PRED, the results of ANN (V_ANN) are close to the experimental ratio and the line of unity, as illustrated in Figures 16 and 17. The scatter of the points for other ratios (CDCs and the CFP) V_EXP/V_ANN are far away from the line of unity. Furthermore, the samples having a value between 20 MPa < f_c< 4 0 MPa and for 1.5% < ρ_l < 2.5% showed the extensive deviation from the line of unity for the ratio V_EXP/V_PRED, as illustrated in Figures 16 and 17. Therefore, there is an urgent need to revise the CDC especially for the mentioned limits of depth, compressive strength (f_c), and longitudinal steel ratio ρ_l (%)

6. Assessment of RC Flat Slab through NLFEA

Mostly the smeared crack and concrete-damaged plasticity (CDP) model are used for studying the nonlinear behavior of concrete in ABAQUS. In the smeared crack concrete modeling as the failure, the cracking phenomenon starts appearing at the failure surface either in biaxial tension or at the tension and compression region [89]. This model is used where concrete cracking occurs due to tension and crushing failure occurs due to compression. When a uniaxial loading testing of concrete is done in tension, it behaves as an elastic material up to 7.5%–11% of final compressive stress after which the specimen starts to crack. This crack formation and propagation occur so quickly that it is very challenging to observe the actual response. As the loss of elastic stiffness gives rise to crack opening and crack is considered as damage to the structure, the model also takes the cracking phenomenon as damage. It is also supposed that there is no permanent sign of the strain associated with the cracking phenomenon which is why crack will be considered as completely close when stresses across the crack become compressive [90].

6.1. Concrete Damage Plasticity (CDP) Model

In the inelastic range, the material behavior of concrete is defined by using the concrete-damaged plasticity model (CDPM) in Abaqus. This model was developed by Lee and Han in 1998. Concrete behavior under low confining pressure is brittle, and using this model, its failure mechanism is based upon stress and inelastic strain [91]. K_c is a ratio of distances between the hydrostatic axis to the corresponding compression and tension meridians. In a deviatory cross section, its value is always higher than 0.5, and when it turns into 1, the failure surface becomes a circle in the deviatory cross section (Drucker–Prager). Similarly, eccentricity is a ratio of tensile strength to compressive strength and according to the Drucker–Prager strength hypothesis, and when eccentricity is zero, the surface becomes a straight line in a meridional plane. To describe the state of the material in the CDP model, another parameter is stress ratio σ_bo/σ_co which may be defined as the ratio of biaxial to uniaxial strength. The dilation angle also affects the performance of concrete under compound stress, which may be defined as the failure of concrete [92]. The default values of the above-discussed parameters are given in Table 3.

To study the nonlinear behavior of slab elements, well-known commercial software ABAQUS was used. First of all, the full-scale sample of the slab was analyzed but it was taking too much time to complete the analysis. To reduce the analysis time, a quarter slab sample was generated and boundary condition was defined to fulfill the continuity condition, as shown in Figure 6. This quarter slab was calibrated against the geometric and material parameters, such as dilation angle, mesh size, viscosity, eccentricity, a ratio of uniaxial stress to biaxial stress, and different element types of steel and concrete. By using the load-displacement technique, the specimen was analyzed and the result was extracted to plot the load-deflection curve. After calibrating the control model, this calibrated model was employed for the remaining specimens and load-deflection curves were plotted with help of MATLAB.

The stiffness degradation of concrete on the strain-softening portion of the stress-strain curve is defined by the two different variables, dt and dc, whose values may vary from zero to one. Zero shows that there is no damage to the material, and one shows a total loss of strength [93]. E_o is the elastic stiffness of the material initially, and , , , and are compressive inelastic strain, tensile inelastic strain, compressive plastic strain, and tensile plastic strain, respectively. The relationship between stress and strain under uniaxial tension and compression is considered as given in following equations:

The strain-softening behaviour of the cracked concrete can be simulated using tension stiffening. Thus, it is required to define the tension stiffening behaviour in the DPMO model. In ABAQUS, we can specify the tension stiffening by the post-failure stress-strain relationship of the concrete or by applying a fracture energy cracking criterion [93].

The variable is associated with crushing and microcracking process within the concrete, and it represents tensile or compressive equivalent plastic strains. Important parameters that need to be defined in the plastic behaviour of concrete using CDPM are the dilation angle ψ_s, viscosity parameter , uniaxial stress to biaxial stress ratio , load eccentricity , and shape factor of the yielding surface (K_c), as discussed in Table 4. The accurate simulation of the tensile behaviour of concrete for the flexural members is very important to achieve accurate results. This model presents a stress-strain model at postfailure conditions of concrete by taking into consideration the tension stiffening behaviour, strain-softening behaviour, and interactions between the concrete and steel bars, as illustrated in Figure 18.

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6.2. Simulations of Steel Reinforcing Bars

In the FEA, steel bars were taken as the elastoplastic material and their interaction with concrete was defined as an embedded region. Similarly, concrete was taken as a host region. A perfect bond between steel and concrete was assumed. Material properties of steel bars used in the present work are yield strength of 420 MPa, Poisson’s ratio of 0.3, and modulus of elasticity as 200 GPa, as illustrated in Figure 19. To simulate the steel bars, a truss element having two nodes and three translations at every node (T3D2) was considered.

For the validation purposes of ANN, selected cases of RC flat slab from the database are modeled in ABAQUS [93], and the information of these cases is summarized in Table 3. Figure 20 described the comparative study of CDCs, ANN, CFP, and ABAQUS [93].

Figure 21(a) illustrates that ABAQUS load-deflection (LD) curves (shown in dotted lines) are in reasonable relation with the experimental curves (shown in solid lines). Figure 21(b) explains the transfer of principal stress, i.e., S33, which is similar to the CFP arch-like frame, as shown in Figure 5, and Figure 21(c) illustrates the plastic strain magnitude (i.e., PEMAG) along the “critical lengths.”

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7. Conclusions

In the current study, capability of ANN modeling is described for RC flat slab with the simply-supported condition under punching load, while the parameters considered are column dimension, c_s, depth of the slab, d_s, shear span ratio, a_vs/d_s, longitudinal percentage steel ratio, ρ_ls, yield strength of longitudinal steel, f_yls, compressive strength of concrete, f_cs, and ultimate load-carrying capacity, V_us.. Following are the main conclusions of the present work:(1)As a result of the comparative study, it has been revealed that nonconventional ANN models predicted the experimental values well compared to conventional models i.e., CDCs and CFP. Similar results have been provided by other researchers using different databases [11, 40–48]. Therefore, behavior of reinforced concrete members with different geometry and loading conditions can better be estimated using ANN models.(2)After training ANNs using a certain database, corresponding structural response can accurately be predicted at ULR irrespective of material behavior and response. From the comparative study, it was revealed that ultimate response of RC members as predicted by ANN models was quite close to the one obtained from EC-2 and CFP and accurate compared to the one obtained from ACI Code.(3)Among the different ANNs proposed, the optimized model was selected based on the R value. For optimized ANN, the R value was found to be 98% and SCS as 4 (c/d, , ρ_l, and f_c/f_y).(4)The estimation obtained from ABAQUS (e.g., NLFEA) of selected case studies shows positive consensus with the experimental results and values provided by ANN. In addition, the numerical predictions obtained seem to support the fact that basic equipment of the RC flat slab specimens investigated is consistent with the visual model adopted by the CFP method.

Data Availability

The data used to support the findings of the study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2021 Afaq Ahmad et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Mathematical Problems in Engineering

Artificial Intelligence for Civil Engineering 2021

Knowledge-Based Prediction of Load-Carrying Capacity of RC Flat Slab through Neural Network and FEM

Abstract

1. Introduction

2. Physical Models for RC Flat Slab

3. Artificial Neural Network (ANN)

3.1. Database of SCS for ANN Modelling

3.2. Preprocessing Phase

3.3. Optimized ANN Model

4. Comparative Studies of Flat Slab

5. Parametric Studies of Flat Slab

5.1. Effective Depth (ds) and Shear-to-Depth against Ratios (avs/ds)

5.2. Compressive Strength (fc) and Longitudinal Steel Ratio ρl (%)

6. Assessment of RC Flat Slab through NLFEA

6.1. Concrete Damage Plasticity (CDP) Model

6.2. Simulations of Steel Reinforcing Bars

7. Conclusions

Data Availability

Conflicts of Interest

References

Copyright

5.1. Effective Depth (d_s) and Shear-to-Depth against Ratios (a_vs/d_s)

5.2. Compressive Strength (f_c) and Longitudinal Steel Ratio ρ_l (%)