Failure-mode–independent prediction model for the peak strength of reinforced concrete columns using Bayesian neural network: A probabilistic approach

A reasonable prediction for the peak strength of reinforced concrete (RC) columns is paramount for the seismic performance evaluation of RC structures. The available prediction models are commonly dependent on the failure mode, and each of them is only applicable to the columns with a particular one. However, the failure mode of RC columns is difficult to be identified accurately in prior, leading to the inconvenience of predicting its peak strength. To overcome this shortcoming, a probabilistic approach was proposed using Bayesian neural network (BNN) to develop a failure-mode–independent model for predicting the peak strength of RC columns directly. The results indicated that the developed model produces reasonable prediction for the peak strength of RC columns failing in different modes. For the training subset, the mean prediction accuracy of the flexure-dominated, flexure-shear-dominated, and shear-dominated columns is 0.997, 0.997, and 0.998, respectively. For the testing subset, the corresponding mean prediction accuracy is 0.957, 0.952, and 0.943. Compared to existing probabilistic models, the developed model exhibits better performance in reducing the uncertainties in peak strength prediction. Compared to existing deterministic models, the developed model could predict the peak strength of RC columns in terms of the confidence interval. In particular, if the confidence interval of peak strength is defined as the mean plus and minus two times standard deviation, 98.9% and 98.4% of the training subset and testing subset are covered. Therefore, the developed model is beneficial for engineers to address the confusion, namely, which peak prediction is the most probable one, when several deterministic models exist for a specific specimen.

Keywords: Experimental database; modeling; peak strength; prediction; reinforced concrete column

Reinforced concrete (RC) columns are probably one of the most critical components of existing RC structures to resist vertical and lateral loads. The post-earthquake observation revealed that RC columns may suffer three different failure modes, namely, flexure failure (FF) mode, shear failure (SF) mode, and flexure-shear (FS) failure mode ([28]; [39]). The above three failure modes exhibit different damage characteristics. In particular, the flexure failure mode performs with ductile behavior with a clear warning before RC columns lose its load-carrying capacity. The other two failure modes exhibit brittle and sudden failure. In seismic design of RC structures, the flexure failure mode is preferable. The other two brittle failure modes, especially the shear failure mode, are prohibited by structural engineers. In spite of this, the shear failure mode and flexure-shear failure mode cannot be avoided completely. Due to the insufficient transverse reinforcement, poor construction quality and boundary condition variation, the two types of failure modes are frequently observed in post-earthquake observation ([7]; [38]; [44]). Therefore, how to account for the influence of failure modes in capacity prediction of RC columns remains one of the major concerns for structural engineers.

The peak strength of RC columns, which represents the maximum bearing capacity to resist the vertical and lateral load, is an important indicator in seismic capacity assessment of RC structures. Due to the complexity of failure mode evolution, predicting the peak strength of RC columns is generally associated with a particular failure mode. During the past years, numerous models have been proposed for predicting this quantity of importance accurately. For example, the conventional fiber-section numerical model was developed to predict the peak strength of flexure-dominated RC columns ([8]; [22]). The empirical regression model [1]; [8]; [41]), machine learning-based model ([19]; [10], [11], [12]), and physics-based models, that is, strut-and-tie model ([15]; [49]), modified compression field theory ([3]; [43]), and truss-and-arch model ([16]; [37]) were developed to predict the peak strength of RC columns failing in shear and flexure-shear modes. For the conventional fiber-section numerical model, it is widely accepted that the peak strength of RC columns failing in flexure mode can be predicted accurately. For the peak strength prediction of RC columns failing in shear and flexure-shear modes, however, it is still far from satisfaction. Therefore, there is a need to improve the prediction accuracy of shear strength models.

Following the two types of peak strength models, identifying the failure mode of RC columns accurately in prior is also important. If a wrong failure mode is identified, it is difficult to predict the peak strength of RC columns accurately. In order to identify the failure mode of RC columns accurately, two indictors, namely, shear aspect ratio and shear demand to capacity ratio were developed over the past decades. Unfortunately, the two indictors show poor performance ([27]; [26]; [28]). For example, [40] concluded that the shear aspect ratio only produces 51% identification accuracy for the flexural failure mode, 63% for the shear failure mode, and 92% for the flexural-shear failure mode. According to [48], the identification accuracy based on the shear demand to capacity ratio can reach 91% for the flexure failure mode, but only 32% for the shear failure mode and 33% for the flexural-shear failure mode. In order to improve the identification accuracy of failure modes, advanced machine learning (ML) techniques such as random forest (RF) and ensemble machine learning algorithms have received extensive attentions in the recent years ([10]; [14]; [23]; [24], [25]).

The failure mode identification accuracy using the advanced techniques has attained great success. For example, the model generated by the AdaBoost learning algorithm gets a high identification accuracy by 0.96 ([10]). However, whether the failure modes of RC columns should be identified in prior to predict the peak strength of RC columns accurately is questioned by the authors ([19]; [29], [30]). In practice, the failure modes of RC columns are not so physical. They are dependent on numerous factors. For example, [5] used a combination of experimental observation and theoretical analysis to label the "actual" failure modes of RC columns. However, the experimental observation provided by experimenters is usually subjective. The threshold values used in the theoretical analysis are also empirical. Then, different observers may draw different conclusion on the "actual" failure modes of RC columns, despite the specimen has the same damage characteristics. In consideration of this, it is suggested to develop a failure-mode–independent model for the peak strength of RC columns. Compared to the previous studies in identifying the failure mode, the failure-mode–independent peak strength model is more deserving in practical application ([19]; [29], [30]).

Toward this end, the data-driven approach is selected to develop the failure-mode–independent peak strength model for its capability of relating the peak strength of RC columns with the important design variables directly. In the recent years, different data-driven approaches, for example, artificial neural network (ANN), support vector machines (SVM), and decision tree (DT) have been employed in this field. For example, [18] predicted the shear strength of fiber reinforced polymer (FRP) concrete flexural beams using ANN. [17] employed SVM to predict the backbone curve of RC columns. [47] used a hybrid machine learning approach to predict the punching shear capacity of FRP-reinforced concrete slab. However, these models are commonly developed in a deterministic manner ([19]), where uncertainties are not included, resulting in a classical confusion, namely, which prediction is the most probable one, exists for engineers ([31], [32], [33]). To solve this problem, the failure-mode–independent peak strength model is developed in a probabilistic manner. In other words, the peak strength of RC columns is treated as a random variable, whose value is predicted in terms of the probability density function (PDF).

The contents of this study are organized as follows. First, the experimental data of RC columns failing in different modes was collected from literatures. Then, the Bayesian neural network (BNN) was introduced to develop the probabilistic failure-mode–independent model, including the back propagation (BP) neural network and the Bayesian inference theory. Furthermore, the advantages of the developed model were demonstrated by comparing with the experimental data and existing deterministic predictions. Finally, a comprehensive parameter sensitivity analysis was conducted to examine the influence of column design parameters on the confidence interval of peak strength.

The experimental database used to develop the probabilistic model was collected from the PEER center, which consists of 251 specimens ([5]). Table 1 enlists the main design parameters of each specimen. As observed, the experimental database covers a wide range of column design parameters, representing typical columns used in practice. Figure 1 shows the statistical distribution of each column design parameter. As observed, the distribution ranges of column design parameters are as follows:

• 1. Column width: $80\text{mm}\le B\le 914\text{mm}$ .
• 2. Column depth: $80\text{mm}\le H\le 914\text{mm}$ .
• 3. Equivalent cantilever length: $80\text{mm}\le L\le 2335\text{mm}$ .
• 4. Compressive strength of concrete: $16\text{MPa}\le f_c\le 118\text{MPa}$ .
• 5. Yield strength of longitudinal reinforcement: $\text{0 MPa}\le f_y\le 1424\text{MPa}$ .
• 6. Yield strength of transverse reinforcement: $\text{0 MPa}\le f_v\le 587\text{MPa}$ .
• 7. Longitudinal reinforcement ratio: $\text{0}\text{.}0068\le \rho \le \text{0}\text{.}0603$ .
• 8. Transverse reinforcement ratio: $\text{0}\text{.}00007\le {\rho }_v\le \text{0}\text{.}0295$ .
• 9. Axial load: $0\le P\le 8000\text{kN}$ .

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Table 1. Experimental database considered in this study.

1 Note: FF, SF, and FS represent the flexure-dominated, shear-dominated, and flexure-shear-dominated failure modes, respectively.

Graph: Figure 1.Statistical distribution of column design parameters in the complied experimental database.

In the experimental database, noted that the "actual" failure modes of RC columns were identified by [5] as a combination of experimental observation and theoretical analysis. In particular, the columns failing in flexural mode are identified by the experimental observation. For the columns without failing in flexure mode, the absolute maximum effective force (i.e., Fe,max) and the peak strength corresponding to the maximum strain (i.e., 0.004) of the transverse reinforcement (i.e., F0.004) are calculated. If Fe,max < 0.95×F0.004 or the displacement ductility at failure is less than or equal to 2, the columns are considered as the shear-dominated one. Otherwise, the columns are considered as the flexure-shear-dominated one. Based on the above classification, there are 197, 36, and 18 columns failing in flexure, flexure-shear, and shear modes, respectively. It is obvious that the number of RC columns failing in flexure mode is much more than that of RC columns failing in shear and flexural-shear mode. So, the developed model will have a tendency to predict well the peak strength of RC columns failing in flexure mode. Meanwhile, the collected experimental data is relatively limited for the model development due to the experimental testing cost. This will affect the generalization capacity of the developed model to some extents.

Given the peak strength of RC columns is unavailable in the experimental database, each column is first standardized in an equivalent cantilever to reduce the influence of the testing configuration. Then, the envelop curves are extracted from the hysteresis curves in terms of the force-displacement relationship. Based on the obtained envelop curves, the maximum strength of RC columns along the positive and negative directions is identified. As shown in Figure 2, the identified maximum strength is not identical along the two directions, despite the specimen is designed and tested symmetrically. This demonstrates well that the prevailing uncertainties induced by experimental testing, concrete material properties, and measurement error are inevitable. To average such discrepancy along the two directions, the peak strength of RC columns is defined as the average values of the identified maximum strength.

Graph: Figure 2.Maximum strengths along the positive and negative directions for RC columns failing in different modes.

Based on the compiled experimental database, the hold-out method is adopted to categorize the experimental data into two subsets, namely, training subset and testing subset. The training subset is categorized to develop the model, whereas the testing subset is categorized to test the generalization capacity of the developed model. For the validation subset, it is not further separated from the training subset to avoid over-fitting because the BNN is advantageous in this problem by penalizing the highly complex model automatically. In the traditional ML algorithm, the validation subset is defined to validate the model as an early stop technique. This technique trains the model until the prediction error starts to increase for the validation subset. This is tedious and computationally expensive. In particular, separating the validation subset from the training subset is impractical if there are only limited experimental data, resulting in the model training has insufficient data. In this regard, the BNN is extremely applicable for the problem with limited experimental data. For the detailed description about the BNN to avoid over-fitting, interested readers can direct to [20], [21] and [34].

For the optimal size between the training subset and the testing subset, there is no widely accepted criterion at present ([9]). Then, a total of 25% of the whole column specimens (i.e., 63 columns) are selected for the testing subset; while the rest of specimens in the experimental database are categorized into the training subset. Figure 3 shows the empirical PDF of column design parameters estimated by the kernel density estimate for the whole dataset and both subsets. As observed, the empirical PDF of both subsets is similar to that of the whole dataset, indicating that both sub-datasets resemble the whole dataset.

Graph: Figure 3.Comparison of PDF for the whole dataset and both subsets.

Normalizing the input and output parameters is paramount because it can reduce the influence of model parameter magnitude. The normalization is generally dependent on the activation function used in BNN. Specifically, the input parameters are normalized with the following expression $$x_{i,n}=\frac{x_i-x_m}{{\sigma }_i}$$

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where xi,n and xi are the normalized and original magnitude of the input parameters in the training subset, respectively; xm and σi are the mean and standard deviation (SD) of xi, respectively. After the normalization, the input parameters have a zero mean and a unit variance. Similarly, the output parameter is normalized with the following expression $$t_{i,n}=\frac{2\left(t_i-t_{i,\min }\right)}{t_{i,\max }-t_{i,\min }}-1$$

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where ti,n and ti are the normalized and original magnitude of the output parameters in the training subset, respectively; ti,max and ti,min are the maximum and minimum values of ti, respectively. Under such normalization, it is apparent that the output parameter will distribute within −1 and 1.

The BNN was originally proposed by [20], [21] and [34], which applies the conventional Bayesian inference in the back propagation (BP) neural network to provide a unified theoretical treatment of the learning in neural networks. Then, the BNN is a combination of BP neural network and Bayesian inference theory. Such design can combine the advantages of ANN and stochastic modeling to represent well the uncertainties in prediction.

In the BNN, the BP neural network is the core element. Following the classical ANN algorithm, the BP neural network is composed of an input layer, several hidden layers and an output layer. In this study, the input layers are the main design parameters of RC columns. The hidden layer includes several neurons which receives the inputs from the previous layer, and then provides an output for the next layer by processing the input data. The output layer contains one or more processing units that produce the output of the network. Obviously, the structure of the hidden layer is unknown and it is dependent on the complexity of the target problem. If the structure of the hidden layer is too simple, the underlying trend of the experimental data cannot be adequately captured. Then, the generalization ability of the developed model is damaged. However, if the structure of the hidden layer is too complicated, the over-fitting may occur and the generalization ability of the developed model is also poor. Therefore, the trial-and-error method is commonly employed for the BP neural network to search for an optimal structure of the hidden layer by changing the number of hidden layers, the number of neurons and the type of activation functions, respectively.

Figure 4 shows the typical BP neural network used in this study. As observed, the established network includes nine neurons in the input layer, ten neurons in the hidden layer and one neuron in the output layer to predict the peak strength of RC columns. In mathematics, the established network relates the input parameters with the output parameter with the following expression $$y_i=f_2〈{\displaystyle \sum \left\{{\omega }_{ij}^{\left(2\right)}\left[f_1\left({\displaystyle \sum {\omega }_{jk}^{\left(1\right)}{x}_k+b_j^{\left(1\right)}}\right)\right]\right\}+b_i^{\left(2\right)}}〉$$

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where xk is the input parameters, yi is the output parameter, and f1 and f2 are the activation functions of the hidden and output neurons, respectively; b and ω are the biases and weights of the network, respectively. To represent the whole parameter space, b and ω are denoted as w.

Graph: Figure 4.Architecture of the typical BP neural network used in this study.

Following the BP algorithm, an optimization algorithm is applied to adjust the magnitude of w in iteration by minimizing the error of the network. For the procedures of adjusting the magnitude of w, interested readers can refer to [6] and [35]. As the error of the network is minimized, the propagation is terminated. In the BP neural network, the objective function is defined as the sum of the squares error between the experimental data and the predicted results, writing $$E_D\left(w\right)=\frac{1}{2}{\displaystyle \sum _{i=1}^N{\left(y_i-t_i\right)}^2}$$

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where ti is the experimental peak strength and N is the number of the experimental data. For equation (4), a complicated weight function may result in a tendency to over-fit the experimental data. Therefore, a regularization technique is commonly applied to modify the objective function by a weight decay term Ew, given as $$S\left(w\right)=E_D\left(w\right)+\alpha E_W\left(w\right)$$

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where α is the regularization coefficient, and $$E_W\left(w\right)=\frac{1}{2}{\displaystyle \sum _{i=1}^m{\omega }_i^2}$$

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where m is the number of the weight parameters. For equation (6), it is obvious that the magnitude of α is unknown and if the magnitude of α is too large, the over-fitting occurs. If the magnitude of α is too small, the developed model will fit the trend of the experimental data inadequately.

In the BNN, the Bayesian inference theory is introduced to treat w as a random variable rather than a deterministic one. In introducing the Bayesian inference theory for the established network, the objective function of the network should be modified by adding another regularization parameter, yielding $$S\left(w\right)=\beta E_D\left(w\right)+\alpha E_W\left(w\right)$$

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where β is the regularization coefficient to express the noise of the experimental data and α is the regularization coefficient to control the weight distribution of the developed model. Then, the posterior distribution of w can be determined by the Bayesian inference theory as $$p\left(w|D,\alpha ,\beta \right)=\frac{p\left(D|w,\alpha ,\beta \right)p\left(w|\alpha ,\beta \right)}{p\left(D|\alpha ,\beta \right)}$$

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where D is the experimental database; p(w|a, β) is the prior distribution of w describing the knowledge of the weight before observing the experimental database; p(D|w, a, β) is the likelihood function of the network; and p(D|a, β) is a normalized factor to assure that the total probability of p(w|D, a, β) is equal to the unity. In the Bayesian inference theory, the normalized factor is also known as the evidence, having $$p\left(D|\alpha ,\beta \right)={\displaystyle \int p\left(D|w,\alpha ,\beta \right)p\left(w|\alpha ,\beta \right)}$$

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It is obvious that evaluating the posterior distribution of w is to determine the prior distribution of w and the likelihood function of the network. According to the investigation by [20], using a normal probability density function (PDF) as the prior distribution of w will greatly simplify the model development. Thus, a normal PDF with a zero mean and a standard deviation of 1/a is applied in this study for the prior distribution of w, expressing $$p\left(w|\alpha ,\beta \right)=\frac{1}{{\left(2\pi /\alpha \right)}^{m/2}}\times \exp \left(-\alpha E_w\right)$$

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For the likelihood function of the network, the model prediction is expressed in PDF, so the probabilistic distribution of the prediction error is defined. Similar to the prior distribution of w, the probabilistic distribution of the prediction error can be described by a normal distribution with a zero mean and a standard deviation of 1/β ([20]), writing $$p\left(D|w,\alpha ,\beta \right)=\frac{1}{{\left(2\pi /\beta \right)}^{N/2}}\times \exp \left(-\beta E_D\right)$$

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Substituting equations (10) and (11) into equation (8), the posterior distribution of w is rewritten $$p\left(w|D,\alpha ,\beta \right)=\frac{1}{{\left(2\pi /\alpha \right)}^{m/2}{\left(2\pi /\beta \right)}^{N/2}}\times \frac{1}{p\left(D|\alpha ,\beta \right)}\times \exp \left[-S\left(w\right)\right]$$

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Then the next step is to find the maximization of equation (12). It is noted that the evidence of the network is independent on the weight and bias. So, the maximization of equation (12) with respect to w is equivalent to the maximization of S(w) with respect to w. Unfortunately, obtaining the analytical solution of p(D | a, β) in equation (12) is intractable. So alternative methods such as Monte Carlo Markov Chain (MCMC) sampling ([2]) or evidence approximation ([4]) are employed. In this study, the evidence approximation is utilized for the capability of obtaining explicit expression. Specifically, S(w) is first rewritten in Taylor expansion around its minimum value. Then, the terms up to the second order are retained to establish the following expression $$S\left(w\right)=S\left(w_m\right)+\frac{1}{2}\mathrm{\Delta }{w}^T{H}_m\mathrm{\Delta }w$$

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where Δw=w-wm and H is the Hessian matrix of S(w), expressed as $$H_m={\nabla }^{2}S\left(w\right)={\beta }^2{\nabla }^2{E}_D+{\alpha }^2{\nabla }^2{E}_W$$

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Submitting equation (13) and equation (14) into equation (12), the posterior distribution of w becomes $$p\left(w|D,\alpha ,\beta \right)=\frac{1}{{\left(2\pi /\alpha \right)}^{m/2}{\left(2\pi /\beta \right)}^{N/2}}\times \frac{1}{p\left(D|\alpha ,\beta \right)}\times \exp \left[-S\left(w_m\right)\right]\times \exp \left[-\frac{1}{2}\mathrm{\Delta }{w}^T{H}_m\mathrm{\Delta }w\right]$$

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According to [20], the posterior distribution of w at its minimum value can be approximated by a normal distribution, having $$p\left(w|D,\alpha ,\beta \right)=\frac{1}{{\left(2\pi \right)}^{m/2}\times {\left({\left|H\right|}^{-1}\right)}^{1/2}\times \exp \left[-S\left(w_m\right)\right]}\times \exp \left[-S\left(w_m\right)\right]\times \exp \left[-\frac{1}{2}\mathrm{\Delta }{w}^T{H}_m\mathrm{\Delta }w\right]$$

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Comparing equation (15) with equation (16), the evidence of the network is obtained by $$p\left(D|\alpha ,\beta \right)=\frac{{\left(2\pi \right)}^{m/2}\times {\left({\left|H_m\right|}^{-1}\right)}^{1/2}\times \exp \left[-S\left(w_m\right)\right]}{{\left(2\pi /\alpha \right)}^{m/2}{\left(2\pi /\beta \right)}^{N/2}}$$

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Differentiating the maximum of equation (17) with respect to a and β, and letting the differentiation equal to zero, finally, the optimal values of a and β are determined. For the detailed description about the differentiation, interested readers can direct to [6].

Figure 5 shows the flowchart of the above stated BNN to develop the probabilistic model for predicting the peak strength of RC columns. As observed, the first step is to draw samples from the prior distribution of w and determine the initial values of a and β randomly. The second step is to train the BP neural network by minimizing S(w) to find the most optimal wm using the scale conjugate gradient algorithm. The third step is to solve equation (15) numerically to obtain the optimal magnitudes of a and β, and the forth step is to repeat the former three steps to obtain the optimal magnitudes of wm.

Graph: Figure 5.Flowchart of incorporating the Bayesian inference theory with the BP neural network.

Following the above stated flowchart, a probabilistic model is developed in this study to predict the peak strength of RC columns failing in different modes. Table 2 enlists the most optimal value of wm after training the developed model. As observed, the established network contains a 9:10:1 architecture, and 111 parameters are optimized in total. Using the obtained most optimal value of wm, the posterior distribution of w is determined by equation (14). Accordingly, the peak strength of RC columns can be predicted in terms of the PDF. To measure its prediction accuracy, a conditional PDF is defined for a specimen with the main design parameters at xi and the experimental data at ti. In particular, the conditional PDF of ti is computed by integrating the whole w space with the following expression $$p\left(t_i|x_i,D,\alpha ,\beta \right)={\displaystyle \int p\left(t_i|x_i,w,D,\alpha ,\beta \right)p\left(w|D,\alpha ,\beta \right)}dw$$

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Table 2. Weights and biases between the input and output layers.

Obviously, equation (18) is difficult to be evaluated analytically. To facilitate the application in practice, the conditional PDF of ti is described by a normal distribution assumption, leading to $$p\left(t_i|x_i,D,\alpha ,\beta \right)=\frac{1}{\sqrt{2\pi {\sigma }^2}}\exp \left[-\frac{{\left(t_i-y_m\right)}^2}{2{\sigma }^2}\right]$$

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where ym is the mean peak strength predicted by the most optimal network weight wm; and σ is the corresponding variance given by $${\sigma }^2=\frac{1}{{\beta }^2}+g^T{A}^{-1}g$$

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where A is the established network and $g={\nabla }_w{y}_m$ is the gradient evaluated at the best estimates. Based on equation (19) and equation (20), it is obvious that the PDF of peak strength can be described by the mean and SD completely. The SD represents the uncertainties in peak strength prediction to produce an error bar on the mean prediction.

Figure 6 shows the mean prediction against the experimental data of RC columns for the training subset and testing subset, respectively. As observed, the coefficients of determination R-squared (R2) are 0.997 and 0.959 for the training subset and testing subset, respectively. Figure 7 shows the mean prediction in comparison with the experimental data for RC columns failing in different modes. As observed, the failure mode does not affect the mean prediction accuracy. For the training subset, the mean prediction of the flexure-dominated, flexure-shear-dominated, and shear-dominated columns has R2 by 0.997, 0.997, and 0.998, respectively. For the testing subset, the corresponding mean prediction has R2 by 0.957, 0.952, and 0.943, respectively. Furthermore, Figure 8 shows the SD of peak strength predicted by the developed model for the training subset and testing subset, respectively. As observed, the SD of peak strength is different when the specimen varies. This is different from the deterministic prediction, which defines a prediction error to represent the uncertainties for all the collected data.

Graph: Figure 6.Comparison of mean prediction and experimental data for (a) training subset and (b) testing subset.

Graph: Figure 7.Comparison of mean prediction and experimental data for RC columns failing in different modes.

Graph: Figure 8.Standard deviation of peak strength for the training subset and testing subset.

Based on the mean and SD of peak strength predicted by the developed model, the confidence interval of peak strength can be defined accordingly. As shown in Figure 9, the confidence interval of peak strength is narrow because the SD of peak strength is less. In spite of this, almost all the experimental data can be covered by the confidence interval of peak strength. In particular, 98.9% and 87.3% of the training subset and testing subset are covered by the confidence interval in terms of the mean ±1 × SD, respectively. If the confidence interval of peak strength is defined by the mean ±2 × SD, 98.9% and 98.4% of the training subset and testing subset are covered, respectively. Compared to existing probabilistic models ([45]; [46]), the developed model performs much better in reducing the uncertainties in peak strength prediction.

Graph: Figure 9.Confidence interval of peak strength for the training subset and testing subset.

Based on the developed model, it is easy to achieve a balance of economic and safety for structural engineers to design RC columns. For example, the peak strength of RC columns can be defined at the mean −2 × SD if the specimen requires a significant conservative design, where the exact value will have a 98% probability to exceed the defined peak strength. If the specimen requires a moderate conservative design, the peak strength of RC columns can be defined at the mean −1 × SD, indicating that the exact value has an 84% probability to exceed the defined peak strength. If engineers wants to have an economic design but with less safety for the specimen, the peak strength of RC columns can be defined at the mean +1 × SD. In this case, the exact value only has 16% probability to exceed the defined peak strength.

To demonstrate the capability of the developed model, the commonly used deterministic peak strength models, e.g., the flexural strength model recommended by, and the shear strength model provided by [1], [8], [41], and [39] are collected from codes and literatures. Then, three specimens are randomly selected from the testing subset as examples, respectively, the flexure-dominated specimen BA3 ([13]), the flexure-shear-dominated specimen 4D13RS ([36]), and the shear-dominated specimen CUW ([42]). Figure 10 shows the PDF of peak strength predicted by the developed model against the peak strength predicted by the considered deterministic models for the three selected specimens. As observed, the deterministic prediction becomes a fractile of the PDF of peak strength. For the flexure-dominated specimen BA3, the peak strength predicted by the is 139.7 kN and the experimental data is 130.9 kN. Yet, the peak strength predicted by the developed model is distributed within 84.8 kN and 183.45 kN, and the most probable value is 132.52 kN, which is in good agreement with the experimental data. For the flexure-shear-dominated specimen 4D13RS, the peak strength predicted by [1], [8], [41], and [39] is 87.2 kN, 116.1 kN, 105.8 kN, and 151.6 kN, respectively. Obviously, the peak strength predicted by the considered deterministic models is dispersed, indicating that it is difficult to predict well the peak strength of RC columns failing in flexural-shear mode. For the experimental data, the corresponding value is 106 kN, which is close to the prediction given by [41]. For the developed model, the peak strength is predicted to distribute within 54.4 kN and 148.6 kN, and the most probable value is 99.3 kN, which agrees well with the experimental data. For the specimen CUW, the developed model predicts the peak strength ranging from 220.8 kN to 326.7 kN, and the most probable value is 273.3 kN. The experimental data is 254.2 kN. For the peak strength predicted by [1], [8], [41], and [39], the corresponding value is 151.2 kN, 229.1 kN, 200.75 kN, and 263.2 kN, respectively. Then, [39] produces the most satisfactory result for the specimen CUW. However, its prediction accuracy fails for the specimen 4D13RS. In this regard, it is difficult to examine the prediction accuracy of the considered deterministic models for a specific specimen. A confusion, namely, which prediction is the most probable one, generally exists for structural engineers.

Graph: Figure 10.Probabilistic calibration of deterministic prediction models.

Finally, a parameter sensitivity analysis is conducted to examine the influence of important design parameters on the confidence interval of peak strength. In parameter variation, the magnitude of column design parameters is varied individually one after another within the distribution ranges of the testing subset; while the magnitude of other column design parameters are fixed at the reference values. Without loss of generality, the median of the column design parameters in the testing subset is defined as the reference values, resulting in: $B$ = 315 mm, $H$ = 345 mm, $L$ =1208 mm, $f_c$ = 69.55 MPa, $f_y$ = 712 MPa, $f_v$ = 294 MPa, $\rho$ = 0.0364, ${\rho }_v$ = 0.0128, and $P$ = 2240 kN.

Figure 11 shows the variation of the confidence interval of peak strength in terms of the mean ±1× SD when the magnitude of column design parameters varies. As observed, the confidence interval of peak strength increases with the increase of B and H. The maximum increase amplitude of B and H is 5.6 and 29.1, respectively, for the mean prediction. This is reasonable because a larger cross-section commonly resists larger external force. For the increase of L, it is interesting to note that the confidence interval of peak strength deceases. The maximum decrease amplitude of the mean prediction is 8.4. This is also acceptable because L is an important parameter affecting the seismic behaviors of RC columns. The variation of L is related to the shear aspect ratio, which affects the relationship between the normal stress and the shear stress. In general, columns with a larger L tend to fail in flexure mode and columns with smaller L tend to fail in shear mode. Compared to the flexural-dominated columns, the shear-dominated columns have larger peak strength and lesser deformation ductility when the other parameters are maintained. For the increase of fc, the confidence interval of peak strength increases linearly, confirming the fact that increasing the concrete compressive strength will enhance the strength and toughness. However, the maximum increase amplitude of the mean prediction is only 1.39. For the increase of fy and fv, the confidence interval of peak strength first deceases and then increases. Such phenomenon can be attributed to the complicated competition between the flexural strength and the shear strength. Specifically, fy is a critical parameter affecting the flexural strength and fv is a critical parameter affecting the shear strength. For the increase of $\rho$ , the confidence interval of peak strength increases. For the increase of ${\rho }_v$ , the confidence interval of peak strength first increases and then decreases. These trends are similar to that induced by fy and fv. Similarly, the reason is related to the competition between the flexural strength and shear strength. Specifically, $\rho$ is a critical parameter affecting the flexural strength and ${\rho }_v$ is a critical parameter affecting the shear strength. Herein, the influence induced by the variation of fy, fv, $\rho$ , and ${\rho }_v$ is less. The corresponding variation amplitude of the mean prediction is only 1.25, 1.42, 1.67, and 1.31, respectively. For the increase of P, extensive investigations have verified that the failure mode of RC columns will change from flexure to shear as the applied axial load increase ([26]). Therefore, the confidence interval of peak strength is expected to increase and the maximum increase amplitude is 1.16 for the mean prediction.

Graph: Figure 11.Parameter sensitivity analysis of column design parameters.

According to the above sensitivity analysis results, each of the considered column design parameters is ranked in a descending order. Figure 12 shows the tornado diagram in terms of the mean and SD of peak strength, respectively. As observed, H, L, and B are the most important parameters affecting the mean prediction, and the other design parameters affect the mean prediction slightly. Among them, the least sensitive design parameters are ${\rho }_v$ and fy. For the SD of peak strength, the influence of column design parameters is complicated. Approximately, the variation of fv, H, and $\rho$ contributes more to the standard deviation of peak strength.

Graph: Figure 12.Tornado diagrams of column design parameters for the mean and standard deviation prediction of peak strength.

A probabilistic approach was proposed in this study to develop a probabilistic model for predicting the peak strength of RC columns failing in different modes. According to the investigations, the following conclusions are drawn:

• (1) The developed model can predict the peak strength of RC columns in terms of the confidence interval. If the confidence interval is defined as the mean ±1× SD, 98.9% and 87.3% of the experimental database in the training subset and testing subset are covered, respectively. If the confidence interval is defined by the mean ±2× SD, 98.9% and 98.4% of the experimental database in the training subset and testing subset are covered. Compared to existing probabilistic model, the developed model exhibits better performance to reduce the uncertainties in peak strength prediction.
• (2) The developed model can predict well the mean peak strength of RC columns. For the training subset, the mean prediction accuracy is 0.997, 0.997, and 0.998 for the flexure-dominated, flexure-shear-dominated, and shear-dominated columns, respectively. For the testing subset, the corresponding mean prediction accuracy is 0.957, 0.952, and 0.943, respectively. Therefore, the failure mode has less effect on the mean peak strength prediction of RC columns.
• (3) The developed model can be used to evaluate the prediction accuracy of existing deterministic models in probability. The peak strength of RC columns predicted by the deterministic models becomes a quantile of the PDF of peak strength. This is beneficial for engineers to solve the confusion, namely, which prediction is the most probable one in practice when several deterministic models exist for a specific specimen.
• (4) The increase of B, H, fc, and P leads to a growth of peak strength with a maximum increase amplitude of the mean prediction by 5.6, 29.1, 1.39, and 1.16, respectively. A shorter column specimen with smaller L leads to a larger peak strength, and the maximum decrease amplitude of the mean prediction is 8.4. For the increase of fy, fv, $\rho$ , and ${\rho }_v$ , the variation of peak strength is complicated because there is a complicated competition between flexural strength and shear strength. Among the considered column design parameters, H, B, and L, are the most important parameters affecting the mean peak strength of RC columns, while the variation of fv, H, and $\rho$ contributes more to the standard deviation of peak strength.

Though the developed model shows great potentialities for the peak strength prediction of RC columns failing in different modes. There are also some limitations and future research directions. First, the problem itself is mainly concentrated on the peak strength of RC columns. There can be more potential possibilities for the problem without clear mechanical interpretation and the case with a large scatter prediction, for example, biaxial stiffness, deformation ductility, and torsional strength. Second, the number of the experimental data is limited due to the experimental testing cost. How to improve the generalization capacity of the developed model using the limited experimental data is still a universal issue in this field. Third, the developed model remains a black box in the context of data-driven approach. For the problem that has not been well solved, how to help the development of mechanical-based model will increase the persuasiveness of the developed model for professional structural engineers.

Chao-Lie Ning https://orcid.org/0000-0001-5643-1414

• B

• = Column sectional width

• H

• = Depth of cross-section

• L

• = Equivalent cantilever length
• f c
• = Concrete compressive strength
• f v
• = Yield strength of transverse reinforcement
• f y
• = Yield strength of longitudinal reinforcement
• ρ v
• = Transverse reinforcement ratio
• ρ
• = Longitudinal reinforcement ratio

• P

• = Applied axial load
• F e,max
• = Absolute maximum effective force
• F 0.004
• = Strength at the maximum strain of 0.004
• x i
• = Input parameters
• x i,n
• = Normalized input parameters
• x m
• = mean of xi
• σ i
• = Standard deviation of xi
• t i
• = Experimental peak strength
• t i,n
• = Normalized experimental peak strength
• t i,max
• = Maximum value of ti
• t i,min
• = Minimum value of ti
• y i
• = Output of the network
• y m
• = Mean peak strength prediction
• σ
• = Predicted variation of peak strength

• A

• = Established network
• f 1
• = Activation functions of the hidden neurons
• f 2
• = Activation functions of the output neurons

• b

• = Biases of the network
• ω
• = Weights of the network

• w

• = Parameter space including _Ii_ and ω

• N

• = The number of the experimental data

• m

• = The number of weight parameters
• α
• = Regularization coefficient to control the weight distribution of the developed model
• β
• = Regularization coefficient to express the noise of the experimental data
• p(w | a, β)
• = Prior distribution of w
• p(D | w, a, β)
• = Likelihood function of the network
• p(D | a, β)
• = Normalized factor

• D

• = Experimental database

• S(w)

• = Objective function
• H m
• = Hessian matrix of S(w)

• FF

• = Flexural failure

• FS

• = Flexural-shear failure

• SF

• = Shear failure

• SD

• = Standard deviation