Lateral-torsional buckling capacity assessment of web opening steel girders by artificial neural networks – elastic investigation

Abstract

Bridge girders exposed to aggressive environmental conditions are subject to time-variant changes in resistance. There is therefore a need for evaluation procedures that produce accurate predictions of the load-carrying capacity and reliability of bridge structures to allow rational decisions to be made about repair, rehabilitation and expected life-cycle costs. This study deals with the stability of damaged steel I-beams with web opening subjected to bending loads. A three-dimensional (3D) finite element (FE) model using ABAQUS for the elastic flexural torsional analysis of I-beams has been used to assess the effect of web opening on the lateral buckling moment capacity. Artificial neural network (ANN) approach has been also employed to derive empirical formulae for predicting the lateral-torsional buckling moment capacity of deteriorated steel I-beams with different sizes of rectangular web opening using obtained FE results. It is found out that the proposed formulae can accurately predict residual lateral buckling capacities of doubly-symmetric steel I-beams with rectangular web opening. Hence, the results of this study can be used for better prediction of buckling life of web opening of steel beams by practice engineers.

Keywords

steel I-beams / lateral-torsional buckling / finite element (FE) method / artificial neural network (ANN) approach

Introduction

Buckling analysis is particularly important for steel structures because they are slender systems. Buckling occurs suddenly and causes the system to collapse. Thin-walled beams with open cross-sections are widely used in many engineering applications. Steel I-beams, loaded in plane, are prone to lateral-torsional and local buckling depending on the overall slenderness of the beam and the slendernesses of the flanges and the web. Lateral-torsional buckling is a flexural torsional buckling mode, in which the cross-sections of a beam translate in lateral direction and twist as rigid bodies with no distortion in the web (Fig. 1(a)). In local buckling, the flanges and/or the web buckle locally over short distances without overall lateral deflection and twist of the beam (Fig. 1(b)). Lateral-torsional buckling and lateral buckling are the characteristic buckling modes of the long- and short-span beams, respectively. On the other hand, in intermediate-length beams with stocky flanges and slender webs, the overall lateral deformation of the beam may be accompanied by web distortions, creating a combination of lateral-torsional buckling and lateral buckling, called the lateral-distortional buckling (Fig. (c)) [1-7].

The critical elastic lateral-torsional buckling of I-beams has been first investigated analytically by Timoshenko and Gere [1]. Trahair [3] presented formula for predicting the elastic critical uniform moment of a simply supported doubly symmetric I-beam which is prevented from lateral deflection and twisting but free to rotate laterally and warping. Bradford and Wee [8] tested eight hot-rolled steel I-beams and supported on seats at the ends. Because the top compressive flange is only restrained by the stiffness of the web, the buckling mode is distortional. It was found out that the code predictions given by the limit states Australian AS4100 [9], and the limit states British BS5950 [10] provide conservative solutions the buckling loads in this case of condition. Zirakian and Showkati [11] carried out distortional buckling experiments on simply supported fabricated steel I-beams with central concentrated load and an effective lateral brace at the mid-span of the top compression flange. The experimental beam strengths were compared with the design strengths predicted by the American AISC/LRFD [12] and the Australian AS4100 [9] specifications. Generally, the two specifications provide unconservative predictions in the inelastic range of structural response as the beam length decreases and inelastic behavior becomes more intense.

There are several existing steel-girder bridges that those lateral-torsional buckling failure’s mode of girders is dominant. In other word, these type of girders not only are not restrained by concrete slab and supported seat but also those length are in a such way that the local buckling will not occurred before lateral-torsional buckling mode. Therefore, it can be supposed the lateral-torsional buckling mode is failure mode of such members. In severe environment corrosion is a dominant type of bridge deterioration. Corrosion of steel occurs when electrolytes are present on the surface, particularly in places where water and contaminants can accumulate. Bridges field inspection show the places most commonly found with corrosion are the top surface of the bottom flange where water collects from dew or splash (Figs. 2) and on the web near the abutments and joints [13-17]. It has been indicated by Kayser and Nowak [17] that severe corrosion may take place at the bottom quarter of the web. Therefore the top surface of the bottom flange and the bottom part of the web are the regions where severe corrosion may take place, as shown in Figs. 2. Corrosion also takes place in the top flange and the top part of the web but the loss is much less as compared to that of the web’s bottom part. Corrosion usually consists of thinning sections in the web or irregularly shaped holes in the web just above the flange and may decrease the load carrying capacity in shear, bearing, and sometimes bending. Therefore, Loss of thickness in web of corroded steel beams which may leads to perforation along corroded regions is one of the main causes of deterioration in aging steel girder bridges in corrosive environment. In addition when used as floor joists, the I-beams sections require web opening to provide access for inspections and various services. The result is a reduction in both the load carrying capacity of girder and level of certainty concerning what the capacity may be. Therefore, it is essential to predict the residual strength of damaged steel bridge girders to investigate the safety level and the retrofitting and rehabilitation strategies in due course.

Previous studies on the lateral-torsional buckling of steel beams with web opening has mainly concentrated on castellated beams or ones with perforation on the center of webs. Redwood and Uenoya [18] have treated the problem of webs as a stability problem of a perforated plate with simplified edge loadings and support conditions. Coull and Alvarez [19] based on their experimental studies, have proposed an empirical method for determining the lateral buckling capacity of beams with a number of openings, either circular or rectangular. Kerdal and Nethercot and Nethercot and Kerdal [20,21] investigate the behavior and stability of castellated beams. In which they provided quantitative data on the lateral-torsional buckling strength of castellated sections, and the similarity in behavior of castellated and plain-webbed beams was shown. The results of their study showed that web opening have negligible influence on the overall lateral–torsional buckling behavior of these beams. Thevendran and Shanmugam [22,23] and Shanmugam and Thevendran [24] proposed a numerical method to predict the elastic lateral buckling load of narrow rectangular and I-beams containing web opening and subjected to single concentrated load applied at the centroid of the cross section using the principle of minimum total potential energy. In another study by Mohebkhah [25] the numerical procedure have been conducted to investigate the inelastic lateral-torsional buckling behavior of castellated beams. Therefore, based on the aforementioned literature it has been considered load-carrying capacity of web opening steel girders using approximated methods specially energy based approach or 2D FEM so far. These methods based approach will not give exact results and cannot estimate the real behavior of deteriorated steel structures. Moreover, the experimental tests suffer financial support to obtain sufficient specimens test. Therefore, the best and appropriate way to consider the load-carrying capacity of deteriorated steel structures is FEM based approach in reasons of the accuracy results and its possibility to have a lot of test results with low cost compare to experimental test.

At present, there is no accepted design method for doubly-symmetric steel I-beams flexural members with different sizes of opening at bottom zone of the web. In the present study, the worst state of loading and opening location in case of lateral-torsional buckling failure mode has been applied on the specimens to assess a conservative prediction for all of damaged steel girders with each loading and boundary conditions. Therefore, it has been supposed that the specimens are subjected to uniform bending along with the midspan opening. Accordingly, a large number of FEA have been carried out on the steel I-beams with single rectangular hole at the bottom zone of web. Then, ANN method is applied using the results of FE analyses to derive new empirical formulae for better predicting the residual lateral-torsional buckling capacity of damaged steel I-beams with rectangular web opening.

Analytical elastic lateral-torsional buckling study

A perfectly straight elastic I-beam which is loaded by equal and opposite end moments is shown in Fig. 3. The beam is simply supported at its ends so that lateral deflection and twist rotation are prevented, while the flange ends are free to rotate in horizontal planes so that the beam ends are free to warp. The beam will buckle at a moment M_{yz} when a deflected and twisted equilibrium position, such as that shown in Fig. 3, is possible. A beam with a uniform moment diagram is the most severe case in the lateral-torsional buckling consideration. The elastic flexural torsional buckling moment of the simply supported doubly symmetric beam in uniform bending is given as below [1,2]:

(1)M_{\mathrm{c}\mathrm{r}}=M_{yz}={\displaystyle \frac{\mathrm{\pi }}{L}}\sqrt{E{I}_y\left(GJ+{\displaystyle \frac{{\mathrm{\pi }}^{\mathrm{2}}}{L^{\mathrm{2}}}}E{I}_{\mathrm{W}}\right)}.

Where E{I}_y is the minor axis flexural rigidity, GJ is the torsional rigidity, and E{I}_{\mathrm{W}} is the warping rigidity of the beam, and L is the length of the beam. In a short-span beam, yielding occurs before the ultimate moment is reached, and significant portions of the beams are inelastic when buckling commences. The effective rigidities of these inelastic portions are reduced by yielding, and consequently, the buckling moment is also reduced [5].

Based on the AISC [26] specification the nominal elastic lateral torsional buckling moment (L_{\mathrm{b}}>L_{\mathrm{r}}), M_{\mathrm{c}\mathrm{r}} for doubly symmetric I-section members has been predicted as:

(2)M_{\mathrm{c}\mathrm{r}}=F_{\mathrm{c}\mathrm{r}}\times S_x{L}_{\mathrm{b}}>L_{\mathrm{r}},

where S_x = elastic section modulus taken about the x-axis; L_{\mathrm{b}} = length between points that are either braced against lateral displacement of the compression flange or braced against twist of the cross section in mm.

(3)F_{\mathrm{c}\mathrm{r}}={\displaystyle \frac{C_{\mathrm{b}}{\mathrm{\pi }}^{\mathrm{2}}E}{{\left({\displaystyle \frac{L_{\mathrm{b}}}{r_{\mathrm{t}\mathrm{s}}}}\right)}^{\mathrm{2}}}}\sqrt{\mathrm{1}+\mathrm{0.078}{\displaystyle \frac{JC}{S_x\times h_o}}{\left({\displaystyle \frac{L_{\mathrm{b}}}{r_{\mathrm{t}\mathrm{s}}}}\right)}^{\mathrm{2}}},

(4)L_{\mathrm{r}}=\mathrm{1.95}{r}_{\mathrm{t}\mathrm{s}}{\displaystyle \frac{E}{\mathrm{0.7}{F}_{\mathrm{y}}}}\sqrt{{\displaystyle \frac{JC}{S_x{h}_o}}+\sqrt{{\left({\displaystyle \frac{JC}{S_x{h}_o}}\right)}^{\mathrm{2}}+\mathrm{6.76}{\left({\displaystyle \frac{\mathrm{0.7}{F}_{\mathrm{y}}}{E}}\right)}^{\mathrm{2}}}},

(5)r_{\mathrm{t}\mathrm{s}}^{}={\displaystyle \frac{\sqrt{I_y{C}_{\mathrm{b}}}}{S_X}},

where C = 1 (for doubly symmetric I-shapes); h_o = distance between the flange centroids; C_{\mathrm{b}} = moment-gradient factor for lateral-torsional buckling; I_y = moment of inertia about the minor axis; E = modulus of elasticity of beam steel; F_{\mathrm{y}} = specified minimum yield stress of the type of steel; J = Torsional constant.

Equation (2) provides identical solutions to Eq. (1) for lateral-torsional buckling of doubly symmetric sections. The advantage of Eq. (2) is that the form is very similar to the expression for lateral-torsional buckling of singly symmetric sections.

Finite element buckling analysis simulation procedure

A powerful commercial structural analysis software, ABAQUS [27], has been employed to simulation in this study. An eigenvalue analysis was used to get the deflected shape and the associated load factor. In an eigenvalue buckling problem it is desired to find the loads for which the model stiffness matrix becomes singular, so that the problem has nontrivial solution. A simple procedure to assess the buckling loads by software can be presented as below:

(6)[k]\{v\}=\mathrm{0},

where [k] is the tangent stiffness matrix when the loads are applied, and the \{v\} is nontrivial displacement solutions. Eigenvalue buckling is generally used to estimate the critical buckling loads of stiff structures (classical eigenvalue buckling). However, even when the response of a structure is nonlinear before collapse, a general eigenvalue buckling analysis can provide useful estimates of collapse mode shapes. ABAQUS can extract eigenvalues and eigenvectors for symmetric matrices only. If the matrices have significant unsymmetric parts, the eigenproblem may not be exactly what the user expected to solve [27].

Mesh and material property considerations

A 3D finite element model is developed to simulate the behavior of doubly-symmetric steel I-beams. An IPE400 section which fabricated from the hot-rolled profile in accordance with the German so-called Estahl Standard was chosen for this investigation. The beam was modeled as three plates representing the two flanges and the web. Four-node thin shell elements (S4R5) were used to model the web and flanges. The flanges are modeled with 6 shell elements over the width and 334 elements along the beam length. The web is modeled with 12 shell elements over the height and 334 elements along the beam length. Figure 4 shows a selected picture of the ABAQUS FE model using the four-node shell elements. Numerical computations are conducted for steel beams that are assumed to be constructed of linear elastic material. A typical value for the modulus of elasticity, E = 210,000 MPa, and Poisson’s ratio was set to 0.3 in this study.

Supports and load conditions

Herein, the girders are considered to be simply supported in flexure and torsion for verification of the developed finite element (FE) program. The flexural and torsional boundary conditions corresponding to simple supports can be modeled as below [27]:

(7)\mathrm{a}\mathrm{t} Z=0,L\to u=v={\displaystyle \frac{\mathrm{d}w}{\mathrm{d}z}}={\displaystyle \frac{{\mathrm{d}}^{2}u}{\mathrm{d}{z}^2}}={\displaystyle \frac{{\mathrm{d}}^2\beta }{\mathrm{d}{z}^2}}=0, \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{a}\mathrm{t} Z=0\to W=0,

where u, v and w are the displacements in the x, y and z directions and β is the angle of twist about the z axis as shown in Fig. 5. The boundary conditions of the analysis models in ABAQUS software are illustrated in Fig. 6. Point A is the hinged end where the displacements in directions 1, 2, 3 and the rotation about direction 3 are restrained. Point B is a roller end where the displacements in directions 1, 2 and the rotation about direction 3 are restrained. To model simple supports at each end of the beam, the central web-node vertical displacements (U_y) and out-of-plane displacements (U_x) are restrained. Also one end only is restrained against longitudinal displacements (U_z). The rotation about the z-axes is restrained at both ends. All degrees of freedom are released, except x-axis translation for both ends thus establishing lateral and torsional supports at both ends. The boundary conditions of the beam are shown in Fig. 6. The end moment loading is simulated with couple forces in the form of uniform loading (w) applied at the top and bottom flanges of the beam end sections with the web height h_w and flange thickness t_f. The top flange is subjected to compressive longitudinal forces, and the bottom flange is subjected to tensile forces opposing the compressive ones as shown in Fig. 7.

Validation of the modeling technique

In this part the accuracy of the finite element model of the beams are investigated. A buckling analysis was run for the model with end moments and the predicted buckling moment was compared with the theoretical value of the lateral-torsional buckling moment capacity. The verification phase starts with evaluating the critical buckling moment of simply supported beams without hole and subjected to uniform bending. The result of the quantitative comparison between analytical solutions and finite element predictions is with an absolute maximum relative error that does not exceed 2.88%. A graphical representation of the beam in the deformed and undeformed configurations is shown in Fig. 8.

Parametric study

After validating the FE model, it has been decided to demonstrate the lateral-torsional buckling capacity reduction characteristics of a beam with web opening under uniform bending by ABAQUS finite element (FE) analysis. The strength of beam with web opening is determined by an eigenvalue analysis, changing geometrical features of holes, which allows for identification of the effect of the single rectangular web opening on lateral-torsional buckling capacity of the beams.

In all, 20 one tests were performed. As stated previously, tested beam is selected from the hot-rolled “IPE400” profiles in accordance with the German so-called Estahl Standard. The specimens were designed 10000 mm lengths and different sizes of opening. Using specifications AISC-LRFD provisions for the beam IPE400 it has been concluded that: L_P = 2121 mm, L_r = 9800 mm, M_p = 291.55 kN·m. Therefore simply supported I-beam, IPE400 (d = 400 mm, b_f = 180 mm, t_f = 13.5 mm, t_w = 8.6 mm), with L (spans length) be equal to 10000 mm>L_r chosen for this investigation under pure bending that buckle elastically. An example for the beam with web opening is shown in Fig. 9.

Characteristic the ratio of hole length to beam length is assumed as b/L = 0 ‒ 0.5L and the ratio of hole depth to beam depth is assumed as a/h = 0 ‒ 0.25h. The web opening characteristics and the lateral-torsional buckling capacity have been shown in Table 1.

The influence of opening length ratio

For the slender beams, the most important type of failure mode is lateral-torsional buckling under different loads and boundary conditions. Moreover, it will be more important as a destruction occurs in service life. Therefore, the importance of horizontal and vertical sizes of opening will be more investigated in this part. The relationship between the buckling moments and the opening length ratio (b/L) under different opening height ratios (a/h) was depicted in Fig. 10. The slope of residual moment-opening length ratio (b/L) curves is larger when the value of a/h gets larger, denoting that the buckling moment of damaged beam with smaller opening height are bigger. As it can be found from Fig. 10, the differences between the residual strength would be much obviously as the opening height is bigger. As shown in Fig. 10, the residual moment-opening length ratio (b/L) curve varies very smoothly. When the value of b/L turns to larger, the curve declines as shown in Fig. 10. The main conclusion is that when the maximum size of opening is occurred the moment capacity will decrease just 10%.

The influence of opening height ratio

The load-carrying capacities of the slender beams are also related to the a/h ratio shown in Fig. 11. The load-carrying capacity of beams with larger opening length ratio is much smaller than that of beams with smaller opening ratio. It can be found from Fig. 11 that the influence of the opening length ratio is more important and sensitive than the opening height ratio. In other words, the length of opening on the reduction strength is more than of the opening height. Therefore, it is desired to do not allow the propagation of opening along the length of beam. Comparison of the Figs. 10 and 11 shows that the effect of opening height is not remarkable and it can be ignored until the 1/4 of cross section height. It can also be found from Fig. 11 that the most strength reduction is occurred in the initial opening height ratio and the strength reduction as increasing the opening height ratio is negligible.

Load carrying capacity of damaged steel beams

Artificial neural network (ANN)

The Artificial Neural Networks (ANNs) are powerful tools for the prediction of nonlinearities using modeling philosophy similar to that used in the development of most of the conventional statistical models. ANNs use the data alone to determine the structure of the model and unknown model parameters instead of using predefined mathematical equations of the relationship between the model inputs and corresponding outputs. They are able to learn and improve as more data become available without repeating the development procedures from the beginning. Hence, they can overcome the considerable limitations of the conventional methods.

An ANN is a mathematical model that emulates biologic neural networks. It consists of interconnected groups of artificial neurons that process information using connectionist approach to computation [28]. It has the ability to learn relationship between input and output provided that sufficient data are available for its training. It does not require an explicit understanding of the mechanism underlying the process, which is the main advantage. The ANN makes use of simple processing units connected by links. The processing unit may be grouped into three main layers namely input layer, hidden layer(s) and output layer. A general Topology or Architecture is presented schematically in Fig. 12. There may be one or more hidden layers before the output layer. Each hidden layer will possess an activation function to compute output to the proceeding layer. The strength of any connection between any two nodes or neurons is provided by weights. Each hidden and output layer processes its input by multiplying each input by its weight and sum the product. Weight may be negative implying that the signal is inhibited by the weight. The sum is further processed using a nonlinear transfer function to produce results. The output of each intermediate hidden layer turns to be input to the following layer. Each processing unit can send out only one output although it normally receives various inputs. The final output produced is compared to the target (actual or desired) output. The weights used for the feed-forward process are adjusted by training the network through data set of inputs and outputs. Furthermore, in ANN models, two sets of data set, named training and testing data sets, are used. The training data set is used to train the network, whereas the testing data set is selected to verify the accuracy of the trained models for the prediction of the remaining strength of damaged beams. Training the neural involves an iterative adjustment of the connection weights so that the network produces the desired output in response to every input signal. Back-propagation network is the most common and powerful technique for training [29], the error produced is systematically distributed backward into the network. Figure 13 illustrate summary of the forward-feed and back-propagation technique of learning /training.

The back-propagation algorithm is used in layered feed forward ANNs. This means that the artificial neurons are organized in layers, and send their signals “forward,” and then the errors are propagated backward [30]. The back propagation algorithm is used for multilayered networks and is a supervised learning process. Supervised learning is the most typically neural network setting. These learning algorithms are characterized by the usage of a given output that is compared to the predicted output and by the adaptation of weights according to this comparison. The back-propagation algorithm is used to minimize the simulation error until the network converges to the expected performance function. Back-propagation looks for the error function minimum in the weight space by applying the gradient descent method [31]. Define the error function for the output of neural network as

(8){ϵ}^{\mathrm{2}}={(traget-OUTPUT)}^{\mathrm{2}},

where ϵ is error, target is dependent variables and OUTPUT is network outputs.

Pu and Mesbahi [32] have proposed a formula for predict ultimate strength of plates using ANN. Hajela and Berke [33] used back-propagation neural network to represent the force displacement relationship in static structural analysis. Ok et al. [34] have proposed formulae that can accurately predict the ultimate strength of locally corroded plates under uniaxial in-plane compression. Guzelbey et al. [35] employed back-propagation neural network for estimation of available rotation capacity of wide flange beams. Fonseca et al. [36] predicted steel beam patch load resistance using back-propagation network. He have also carried out parametric studies based on the neural network model, and furthermore they have proposed a neuro-fuzzy system for the parametric analysis of patch load resistance.

ANNs approach is used to predict the remaining buckling strength of the damaged steel beams in this study. Empirical modeling by heuristic and modern search techniques such as ANNs is a different approach to determine the remaining buckling strength of beams. The training and test models are developed based on a comprehensive database that was obtained from the FEM approach.

Selection of input and output

As noted in previous sections two parameters that should be considered as lateral-torsional buckling capacity reduction factors are x₁ = b/L and x₂ = a/h and location of the web opening. As stated previously the worst state maybe approximated as the web opening occurred in mid-span of beam. Besides, the critical loading and boundary conditions are also as Figs. 6 and 7 respectively. Therefore, the pessimistic conditions of damage, loading and boundary have been considered here to derive a practical formula. In other word this formulae maybe employed for other conditions conservatively. Accordingly the input vector selected for this model is {x₁, x₂}. We would like to know the lateral-torsional buckling capacity of steel I-beams with rectangular web opening for any given problem. x₁ = a/h has five values ranging from 0 to 0.25. x₂ = b/L has 6 values ranging from 0 to 0.5. Accordingly, the output vector for the neural network model is selected as {M_b /M_bo}, where M_bo and M_b are the lateral-torsional buckling subjected to uniform end moment for intact and damaged beams, respectively.

ANN structure

Choosing the appropriate number of hidden neurons and number of hidden layers are major parameters in obtaining an accurate ANN model. In addition, the best selection of activation function has a considerable effect on the ability of the model. The number of hidden layers and number of nodes in hidden layers are usually determined via trial and error procedures or using suggested rules. After doing a few trials, it is observed that the network has one hidden layer because this kind of model has been found to have sufficient accuracy and less demand on the amount of training data. The number of neurons in input layer is equal to the number of variables. There are two neurons in the input layer, e.g., x₁ and x₂. Also the number of neurons in output layer is equal to one. Each input is weighted with an appropriate w. The sum of the weighted inputs and the bias, forms the input to the transfer function. Neurons can use any differentiable transfer function to generate their output. The neural network's configuration has been shown in Fig. 14.

The network that is used for this study is a two-layer feed forward network, with hyperbolic tangent transfer functions in the hidden layer and identity transfer functions in the output layer. The outputs using a logistic activation function can be expressed as

(9){\left({\displaystyle \frac{M_{\mathrm{b}}}{M_{\mathrm{b}\mathrm{o}}}}\right)}_{\mathrm{s}}=\left|W_{\mathrm{2}}\times \left[\begin{array}{l}\mathrm{1}\\ \tanh (W_{\mathrm{1}}\times X)\end{array}\right]\right|,

where,

W_{\mathrm{1}}=\left[\begin{array}{lll}{b}_A& W_{\mathrm{1}A}& W_{\mathrm{2}A}\\ b_B& W_{\mathrm{1}B}& W_{\mathrm{2}B}\end{array}\right],

W_{\mathrm{2}}=\left[\begin{array}{lll}{b}_F& W_{AF}& W_{BF}\end{array}\right],

X={\left[\begin{array}{lll}\mathrm{1}& x_{\mathrm{1}}& x_{\mathrm{2}}\end{array}\right]}^{\mathrm{T}}.

In this study back-propagation algorithm is used in layered feed forward ANNs for training. During the training procedure, the input and desired data will be repeatedly presented to the network. As the network learns more and more, the error tends to drop toward zero. Updates the weights after passing all training data records, it is most useful for small data sets. The maximum number of epochs is equal to 10000. If the maximum number of epochs is exceeded, then training stops.

Proposed formulae to residual lateral-torsional buckling capacity assessment

The derived formula for predicting lateral-torsional buckling capacity of steel I-beams with rectangular web opening is expressed as:

(10){\displaystyle \frac{M_{\mathrm{b}}}{M_{\mathrm{b}\mathrm{o}}}}=\mathrm{0.03104}\mu +\mathrm{0.9708},

where M_b is lateral-torsional buckling capacity of steel I-beams with rectangular web opening.

(11)\mu =\left|W_{\mathrm{2}}\times \left[\begin{array}{l}\mathrm{1}\\ \tanh (W_{\mathrm{1}}\times X_S)\end{array}\right]\right|,

X_S={\left[\begin{array}{lll}\mathrm{1}& X_{S\mathrm{1}}& X_{S\mathrm{2}}\end{array}\right]}^{\mathrm{T}},

X_{S\mathrm{1}}=\mathrm{12.8783}\left({\displaystyle \frac{a}{h}}\right)-\mathrm{1.9563},

X_{S\mathrm{2}}=\mathrm{6.4168}\left({\displaystyle \frac{b}{L}}\right)-\mathrm{1.8333},

W_{\mathrm{1}}=\left[\begin{array}{lll}-\mathrm{1.054}& \mathrm{0.023}& \mathrm{0.985}\\ \mathrm{0.606}& -\mathrm{0.15}& -\mathrm{0.855}\end{array}\right],

W_{\mathrm{2}}=\left[\begin{array}{lll}-\mathrm{1}& -\mathrm{1.312}& \mathrm{0.759}\end{array}\right].

.

Regression analysis of the network outputs and desired outputs (targets) are then carried out to characterize the network accuracy. Figure 15 shows the correlation of both FEM results and ANN outputs.

It is evident from Fig. 15 that neural network is valid for predicting the lateral-torsional buckling capacity of steel I-beams with rectangular web opening.

Conclusions

This paper has presented the details of an investigation into the elastic lateral-torsional buckling behavior of hot-rolled steel I-beams with different sizes of rectangular web opening that happened by severe corrosion. As stated previously, the worst state of loading and opening location has been applied on the specimens to assess a conservative prediction for all of damaged steel girders. In this case the analyzed specimens have been supposed to be subjected to equal and opposite end moments with an opening in midspan. A FE eigenvalue analysis was used to get the buckling strength of beams using commercial analysis software ABAQUS. It was found from the FE results that, the opening length has more effect than opening height on the strength reduction, and the effect of opening height to 1/4 of cross sectional height may be ignored. To predict residual lateral-torsional buckling capacities of damaged steel girders the ANN approach has been also employed. Using the ANN approach an exact formulae has been derived with two independent different inputs consisting of the opening length and height ratios. Hence, the realistic appraisal of the capacity of corrosion damaged beams using these methods will avoid plant closures when the capacity of steel works may be adequate. In addition, these methods may be used to identify the weaker members whose capacities are closer to the service loads. These assessment methods will help the practicing engineer to make a fast and reliable decision regarding the future of corrosion damaged I-beam.

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