Structural members within buildings and civil infrastructure are typically designed considering a pre-defined “serviceable design life,” in accordance with the respective national building codes and standards. Throughout this stipulated design lifetime, the structure will naturally face a variety of scenarios that may affect the intrinsic structural load-carrying capacity, such as the deterioration of the structural members through corrosion (Doebling et al., 1998) or, perhaps, owing to a change in the design requirements that results in additional loading caused by changes in use or increased occupancy (Ross et al., 2016; Askar et al., 2021; Slaughter, 2001). With the construction industry contributing about 30% of greenhouse gas emissions and energy consumption globally (UNEP, 2020), from which a substantial amount of demolition waste is generated (Publications Office of the European Union, 2017), the industry needs to focus on efforts to reduce its impact. The rehabilitation of existing structures is one such effective method to reduce the embodied carbon of a structure, thereby reducing its environmental impact (Alba-Rodríguez et al., 2017). However, rehabilitating a structure for reuse requires an assessment of the in situ health of the constituent structural members and components such that strategies for strengthening and rehabilitation may be developed.

Structural health monitoring (SHM) covers a wide spectrum of techniques, where essentially the response of structures and their individual components to applied actions is recorded to determine their mechanical state and current health (Gharehbaghi et al., 2022; Katam et al., 2023; Amafabia et al., 2017). SHM techniques can be classified into four categories of increasing levels of complexity, namely, “detection,” “localization,” “quantification,” and “prediction of the remaining life” (Rytter, 1993). The SHM techniques can be further distinguished by the nature of the actions applied, where “static-based methods” measure the response of a structure to quasi-static loads (such as stiffness, strains, and stresses) and “dynamic-based methods” measure the structural response to dynamic loading (such as the frequency response to dynamic loading) (Gharehbaghi et al., 2022; Shokravi et al., 2020). In essence, both classes of methods aim to evaluate the current mechanical state, or health, of a structure, given its response to an applied action, be it a static or dynamic perturbation. Static-based methods have been used in civil infrastructure, such as bridges, where parameters such as displacements, strains, and strut and cable stresses can be measured for identifying structural damage (Chen et al., 2016; Martínez et al., 2016; Wu et al., 2018). Dynamic-based methods are also used extensively in the industry for damage identification through the vibrational characteristics of a structure (Koh and Dyke, 2007; Fan and Qiao, 2011; Hakim et al., 2014; Favarelli et al., 2021). Gharehbaghi et al. (2022) noted that the measurements of static responses were much more straightforward than those of dynamic-based responses since the dynamic-based responses require the meticulous management of operational and environmental effects for obtaining accurate data.

Beam–columns are ubiquitous structural members within steel-framed buildings, transferring the vertical and lateral loads acting on the building to the foundation (Lindner, 1997). Their importance in providing stability to the overall frame underscores the critical need for developing a robust, practical, and cost-effective SHM technique as being an essential precursor for extending the design service life of existing structures (Liu and Nayak, 2012; Thomson, 2013; Sumitro and Wang, 2005). For the outcomes of an SHM procedure to achieve this goal, it must be able to determine the “current” (in situ) structural capacity of the structural member or, at least, be able to provide sufficient information to predict its current proximity to failure. The current paper investigates the feasibility of using the probing SHM procedure, introduced by Shen et al. (2023), to evaluate the health of simply supported beam–columns subjected to axial compression and uniform bending.

Considering first the design of slender steel columns under a purely axial load, these are typically governed by buckling instabilities (Timoshenko and Gere, 1963; Allen and Bulson, 1980), where the ultimate axial load capacity ( P ult ) is equal to the Euler buckling load, P E = π 2 E I / L e 2 , where E and I are the Young’s modulus of the material and the second moment of the area of the cross section, respectively, while L e is the effective buckling length of the column that varies with the boundary conditions at the supports. Considering next the design of slender members subjected to uniaxial bending, the ultimate bending moment capacity ( M ult ) is typically governed by lateral torsional buckling (LTB). The effects of LTB can be mitigated by restraining the beam to prohibit the lateral deformation or by using cross sections with high torsional and warping stiffness such as closed sections (Kitipornchai and Trahair, 1980; Trahair et al., 2008). The ultimate bending moment for members that are designed to be laterally restrained is typically governed by plasticity and is, hence, dependent on the cross-sectional geometry and material properties.

The current study considers members that are subjected to a combination of both axial and bending forces simultaneously, wherein the effects of LTB are restrained, and henceforth referred to as “beam–columns.” This is achieved by considering a beam–column with a circular hollow section (CHS). In practice, LTB can also be restrained in beams that support a floor slab since the slab restrains the compression flanges of the beam. Hence, the study of laterally restrained beam–columns represents a more straightforward, yet practically realistic, design scenario faced within the industry; the explicit consideration of laterally unrestrained beam–columns is left for future work. Furthermore, the effects of the local–global buckling interaction are not presently considered. The failure criterion for laterally restrained beam–columns is more intricate since it is determined by the member response to both bending and compression actions (EN, 2014; Liew and Gardner, 2015; Arrayago et al., 2015; Cavajdová and Vican, 2023). This response is best demonstrated by the interaction plot shown in Figure 1, where P ^ is the applied axial compression force P normalised by the ultimate axial capacity of the member P ult , and hence, P ^ = P / P ult , while M ^ is the applied moment M normalised by the ultimate bending capacity of the member M ult , and hence, M ^ = M / M ult . In Figure 1, the solid line indicates the failure boundary envelope, i.e., the combinations of M ^ b and P ^ b that would induce the onset of failure within the member. Hence, combinations of M ^ and P ^ that lie below the curve are safe-sided, as depicted by the coordinate M ^ safe , P ^ safe in Figure 1; conversely, points that lie above the interaction boundary cause failure and are unsafe, as depicted by the coordinate M ^ unsafe , P ^ unsafe . Obtaining M ^ and P ^ for an in situ structure would provide designers with a metric that can inform the potential structural strength reserve of the beam–columns, i.e., addressing the criteria for SHM mentioned above. Previous work has stated that this is a critically important area of research for present SHM methodologies (Gharehbaghi et al., 2022). Consequently, any structural health monitoring index for beam–columns must be evaluated in relation to this boundary, which is discussed in Sections 3,4. Once the results from the probing–machine learning framework are presented and analysed, a brief discussion on the prospects of future developments is presented, and then, conclusions are drawn.

The probing methodology was originally developed by Thompson (2015) to assess the buckling resistance of shells using a non-destructive technique. The process involved loading a cylindrical shell to a prescribed axial compression load and is subsequently probed laterally with a small probing force F p while recording the corresponding probing lateral deflection δ p . The prescribed axial compression load is then increased, once again recording the response to probing, until all the probing responses for each normalised axial compression level are recorded. The results are plotted, which enables the notoriously difficult-to-predict buckling load of the cylindrical shell to be obtained (Shen et al., 2023). The concept of using probing as a monitoring technique was devised by Shen et al. (2023) by considering the case of axially loaded prestressed stayed columns (PSCs). In their study, the normalised axial utilisation ratios, equivalent to P ^ currently, alongside the degree of cable erosion, were successfully inferred from the probing response of the PSC. This methodology, again, is non-destructive, can be executed with minimal interruption, and may be classified as a periodic visit-based monitoring (PVM) technique since continuous monitoring via sensors is not required. The current work extends the developed probing SHM methodology to predict the utilisation ratio of simply supported beam–columns subjected to combined axial and bending loads.

The procedure adopted here is similar to that implemented for the study of PSCs (Shen et al., 2023), where the more generic steel beam–column is modelled within the commercial finite element (FE) analysis software application Abaqus (Dassault Systèmes Simulia Corp, 2021). The modelled beam–columns, under a specified combination of axial force ( P ) and uniform bending moment ( M ) , are probed laterally at a prescribed location L p with a nominal probing force F p ; in the present study, F p = 100   N . The probing response for the beam–columns is then recorded by measuring the corresponding displacement δ p , as shown in Figure 2, with this procedure being repeated for varying combinations of M ^ and P ^ to generate the dataset for the study presented later. Throughout the parametric study, it is observed that the probing response is typically linear, as reported for PSCs by Shen et al. (2023). However, it is noted that certain combinations of M ^ and P ^ considered within the parametric study exhibit a nonlinear response to probing. This is owing to M ult being governed by material plasticity for a laterally restrained beam; hence, probing potentially causes the member to be loaded beyond its elastic limit. A fundamental principle of probing in the structural health assessment methodology is that the structure must remain elastic during the probing process to ensure that any of its effects are transient and reversible (Shen et al., 2023). Therefore, combinations of M ^ and P ^ that undergo permanent deformation through plasticity, presently termed “plastic points,” are considered to be beyond the applicability of the probing procedure and are hence excluded from the study. The probing stiffness k p is evaluated for each loading combination considered using the relationship k p = d F p / d δ p . The evaluated k p is subsequently used as an input for the artificial neural network (ANN) surrogate machine learning (ML) model to infer the utilisation ratios, M ^ and P ^ , of the beam–column; hence, the ANN ML model aims to solve the “inverse” problem to that of the FE analyses. The use of ML surrogate models provides a powerful solution for capturing complex structural behaviours that are difficult to model with traditional methods (Wu et al., 2022; Xing et al., 2023). In theory, with sufficient training, the surrogate ANN ML model should be able to infer both M ^ and P ^ when the beam–column is probed laterally in situ. Details of the surrogate ANN ML model are given in Section 4.

The presence of the in-service bending moment M , along with the axial load P , induces a curvature in the beam–column. The deflected profile of this beam–column can be found by considering the differential equation of equilibrium, given by Timoshenko and Gere (1963): E I w ″ + P w = − M , (1) where w describes the deflected profile as a function of the longitudinal coordinate x , as shown in Figure 7, while primes ( ′ ) denote derivatives with respect to x . The general solution to Equation 1 is known to be of the form w x = C 1 ⁡ sin ⁡ β x + C 2 ⁡ cos ⁡ β x − M P , (2) where C 1 and C 2 are constants that depend on the structural boundary conditions, while β is defined in Equation 3 as: β = P E I . (3)

For a beam–column with uniform bending moment M , the boundary conditions are − E I w ″ 0 = − E I w ″ L = M . (4) Thus, by substituting the conditions in Equation 4 into Equation 2, the constants C 1 and C 2 can be determined, which yields w x = M P 1 − cos ⁡ β L sin ⁡ β L sin ⁡ β x + cos ⁡ β x − 1 . (5)

The deflected profile in Equation 5 is therefore dependent on M and P . Note that as P → 0 , and thereby, β → 0 , the deflection function w converges to a parabola in x , as described by Euler–Bernoulli bending theory.

Owing to the complex behaviour of beam–columns under combined bending and axial loading, the direction of the probing force F p is important. For load combinations that are close to the interaction boundary, probing the beam–column in the positive direction, as shown in Figure 7A, leads to a nonlinear probing response, as shown by the solid line in Figure 8A for the load combination ( M ^ = 0.45 , P ^ = 0.35 ). This behaviour was not observed for load combinations that were remote from the interaction boundary, as shown by the dashed line in Figure 8A, which represents the probing response for load combination ( M ^ = 0.15 , P ^ = 0.05 ). This nonlinear probing response was caused by the positive probing force F p , triggering an inelastic response that violated one of the principal aims of the probing procedure for structural health assessment, i.e., ensuring that the probing intervention must trigger an elastic (hence reversible) response from the structure, as described in Section 2. One potential strategy to avoid triggering an inelastic response is to probe the member in the “negative” direction, i.e., in the opposing direction to the deflected shape, as shown in Figure 7B. This ensures that a linear and elastic probing response of the beam–column is maintained, as shown by the dashed line in Figure 8B, where a linear probing force–displacement response is observed when probing in the “negative” direction. This is currently attributed to the fact that probing in the negative direction is equivalent to applying a negative moment ( − M ) to the member, which, in effect, unloads the member, and hence, the probing equilibrium path follows a linear elastic response.

A simplified and safe method for identifying the aforementioned “plastic points” is to determine the “elastic boundary” by considering the linear addition of the absolute stress blocks from the applied moment M and the axial compression P while limiting the peak stresses to be less than the yield stress of f y . Mathematically, the addition of the absolute stress blocks can be expressed as P A + M z I < f y , (6) where A and I are the cross-sectional area and the second moment of the area of the beam–column, respectively, while z is the distance from the geometrical centroid of the cross section to its extreme fibre. Using the definitions of P ^ and M ^ , respectively, and substituting them into Equation 6, yields Equation 7: P ^ P ult A + M ^ M ult z I < f y , , (7) which can be used to eliminate combinations of M ^ and P ^ that experience plasticity, as shown in Figure 9. It was found that points that lie above the elastic boundary exhibit a nonlinear probing response, confirming the accuracy of the derived boundary. Therefore, the present study considers load combinations that lie within the interaction and elastic boundary to avoid triggering plasticity during probing. Moreover, combinations of M ^ and P ^ that are excessively close to the interaction boundary are also excluded from the analysis. This was achieved by bringing the interaction boundary inwards towards the origin by 0.05 M ^ and 0.05 P ^ ; this is presently termed the “offset boundary.” Therefore, the loading combinations of M ^ and P ^ that lie within these established boundaries are considered here and are indicated by blue dots in Figure 9.

The probing response for all combinations of M ^ and P ^ is considered in the parametric study conducted in Abaqus; the results showing the variation in k p and w ini alongside M ^ and P ^ are shown in Figures 10A, B. A direct correlation between P ^ and k p is readily observed by the linear variation in the P ^ -axis direction in Figure 10A, echoing the findings obtained by Shen et al. (2023). However, a similar correlation between M ^ and k p is not readily found since the response is practically constant for different M ^ values, while P ^ is constant. However, a variation in w ini with both M ^ and P ^ is found, as shown in Figure 10, which strongly suggests that w ini is a suitable parameter to consider in the ML framework.

Using Equation 5, the FE results can be verified against w ( x ) by substituting the probe location into x , with the resulting analytical values shown in Figure 10C. It is shown that the calculated values for w ( x ) agree reasonably well with w ini obtained from Abaqus as long as the response remains elastic. This is shown in Figure 11, which depicts the ratio of the analytical deflection value to that determined from Abaqus, where values equal to unity denote perfect agreement. It is shown in Figure 11 that all the results lie within the range 0.9 , 1.0 , with the vast majority lying in the range 0.99 , 1.00 , which implies that the probing response generated by Abaqus is sufficiently accurate to be utilised as the training data in the ANN ML model to predict M ^ and P ^ .

Following the framework developed by Shen et al. (2023), an ANN is implemented for the machine learning framework in the current study to determine whether this can be used to predict M ^ and P ^ for beam–columns. Here, the ANN is developed using TensorFlow (Abadi et al., 2016) with the high-level application programming interface Keras (Chollet, 2015).

As discussed in Section 3, the probing stiffness response k p alone may be insufficient to determine M ^ and P ^ , owing to the lack of variation in the former with k p . Consequently, it was hypothesised that different input cases are required to generate a good prediction of M ^ and P ^ in the ANN. Hence, the suitability of k p and w ini measured at different probing locations as ANN input parameters is explored in different scenarios, which are outlined in Table 2. Owing to the symmetry of the loading conditions and geometry, probe locations are limited to the lower half of the member, i.e., L p = L / 4 , L / 3 , L / 2 , to avoid duplicating data responses in the ANN. The inputs ( k p , u ini ) are then normalised using the MinMaxScaler function within the sklearn.preprocessing package, which is known to assist in achieving convergence by reducing scaling effects when training the ANN model (Montavon et al., 2012). The total number of data points used varies according to the input cases, as shown in Table 2.

The ANN model predicts two outputs, M ^ pred and P ^ pred , which can be presented as coordinates in the M ^ – P ^ space. Following an initial hyperparameter optimisation study, the structure of the ANN is formed of three hidden layers, each having 20 neurons, which produces sufficiently accurate predictions without excessive computational effort. The activation function used for the neurons in the internal layers is chosen to be the rectified linear unit (ReLU) (Agarap, 2018), while for the output layer, a sigmoid function was used to ensure that M ^ pred and P ^ pred lie in the range 0,1 . The framework for the ANN is shown in Figure 12, which depicts the “70–30” split of the training data, i.e., 70% of the probing results are used as the training dataset, and the remaining 30%, termed the “evaluation dataset,” are set aside as an “unseen and unbiased” dataset to evaluate the accuracy of the trained ANN. From the initial hyperparameter study, the optimum number of epochs was determined to be 2,000.

In the ML framework, loss functions are used to quantify the model error in the training space, and they are used to optimise the weights and bias between each layer in the ANN (Kerkhof et al., 2023). Currently, two types of loss functions are explored, namely, the mean absolute error (MAE) and a bespoke loss function termed “Euclidean distance-loss function” (EDLF) developed by Sung Lee et al. (2020). The difference between the MAE and EDLF is that the MAE computes the average absolute error between the predicted and actual outputs, averaging both the errors of M ^ and P ^ together (Hodson, 2022), whereas EDLF calculates the distance in the Euclidean space between the ML-predicted coordinate points against the training dataset true coordinate points in the M ^ – P ^ two-dimensional space (Sung Lee et al., 2020). The evaluated loss values, calculated using either the MAE or EDLF, are back-propagated into the ANN model (Larochelle et al., 2009) in a manner that minimises subsequent training losses using the Adam optimisation algorithm (Kingma and Ba, 2015). This optimisation procedure is repeated until the training epoch is completed. Subsequently, upon completing the training loop, the test dataset, which was initially set aside, was used to evaluate the performance of the trained ANN model, thus evaluating the model against “unseen” data, thereby preventing potential bias in the results. The results obtained from the evaluation dataset fed into the trained ANN model are then plotted in Figures 13–15.

The probing response of a perfect beam–column that is restrained against the effects of LTB was explored in the current work using the commercial FE software application Abaqus. Owing to a more complex response for probing a beam–column, when compared to a member under pure compression, it is shown that the probing stiffness k p provides sufficient data to predict P ^ , but in isolation, it is insufficient for predicting M ^ since k p is essentially invariant with the applied moment. However, this is remedied by including a measurement of the initial deflection before probing, w ini , which restores accuracy to the developed ANN. Subsequently, the developed ANN is demonstrated to predict M ^ and P ^ accurately when k p and w ini at different probing locations are used as inputs.

The present study also highlights a potential risk associated with the implementation of probing as a structural health assessment tool for beam–columns designed to fail with a plastic ultimate moment of resistance, M ult . To mitigate the risk of damaging the member during the probing process, the member should be probed such that the local deflection is reduced, which would ensure that the probing response of the column remains linear. A further recommendation is that the probing procedure is not conducted within 5% of the elastic limit. Future research will elaborate on the current findings by exploring different scenarios, such as laterally unrestrained beams while also focussing on experimental studies to validate the current findings while ascertaining the practicality of in situ probing as a methodology for assessing structural health.