# Neural network-based analytical model to predict the shear strength of steel girders with a trapezoidal corrugated web

Miguel Abambres, Jun He

► To cite this version:

Miguel Abambres, Jun He. Neural network-based analytical model to predict the shear strength of steel girders with a trapezoidal corrugated web. 2018. hal-02074778v3

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# Neural network-based analytical model to predict the shear strength of steel girders with a trapezoidal corrugated web

Miguel Abambres<sup>a</sup>, Jun He<sup>b</sup>

<sup>a</sup> R&D, Abambres' Lab, 1600-275 Lisbon, Portugal
abambres@netcabo.pt

<sup>b</sup> School of Civil Eng, Changsha University of Science and Technology, 410114, Changsha, PR China
hejun@csust.edu.cn

## Abstract

Corrugated webs are used to increase the shear stability of steel webs of beam-like members and to eliminate the need of transverse stiffeners. Previously developed formulas for predicting the shear strength of trapezoidal corrugated steel webs, along with the corresponding theory, are summarized. An artificial neural network (ANN)-based model is proposed to estimate the shear strength of steel girders with a trapezoidal corrugated web, and under a concentrated load. 210 test results from previous published research were collected into a database according to relevant test specimen parameters in order to feed the simulated ANNs. Seven (geometrical and material) parameters were identified as input variables and the ultimate shear stress at failure was considered the output variable. The proposed ANN-based analytical model yielded maximum and mean relative errors of 0.0% for the 210 points from the database. Moreover, still based on those points, it was illustrated that the ANN-based model clearly outperforms the other existing analytical models, which yield mean errors larger than 13%.

**Keywords:** shear strength; corrugated webs; trapezoidal corrugation; steel girders; artificial neural networks; analytical model

## 1. Introduction

Corrugated steel plates without additional stiffeners are characterized by high shear buckling strength and out-of-plane flexural stiffness, having been widely used in structural engineering applications, such as large span roof, steel plate shear walls, and bridge girders (Wu et al. 2006, He et al. 2012, Emami et al. 2013, Jiang et al. 2015). The concept of replacing
flat webs with corrugated webs in bridge structures was first proposed in France at 1986, then
successfully applied worldwide in the past 30 years (Combaut 1988, Kondo et al. 1994).
Typically, the corrugations in the web are trapezoidal, but forms like sinusoidal, triangular, and
rectangular have also been considered. Corrugations are used to increase the shear stability of
webs in beam-like members, thus eliminating the need of transverse stiffeners. This paper
focuses on the shear strength of steel webs with trapezoidal corrugations, a topic that has been
extensively studied since the end of 20th century, covering both buckling and plasticity
phenomena (e.g., Bergfelt & Leiva-Aravena, 1984, Leiva-Aravena & Edlund 1987, Johnson &
Cafolla 1997, Metwally 1998, Sayed-Ahmed 2001, 2007, Abbas 2003). Shimada (1965) was
the first to study the shear strength of steel girders made of folded-plate webs. Easley and
McFarland (1969) proposed the global shear buckling equation of corrugated webs by treating
them as flat and orthotropic. Lindner & Aschinger (1988) carried out experimental tests to
assess the shear strength of trapezoidal corrugated steel webs, and suggested using 70% of the
elastic shear buckling stress as the nominal strength for design purposes. Luo & Edlund (1994,
1996) analysed the buckling of trapezoidal corrugated panels under in-plane loading by spline
finite strip and finite element (FE) methods. The influence on the elastic buckling load of
various parameters, such as geometry, loading patterns and boundary conditions, was assessed.
Elgaaly et al. (1996) presented experimental and analytical results for steel beams with
trapezoidal corrugated webs loaded predominantly in shear, and proposed bucking formulas
based on (i) the local deformation of the corrugation folds modelled as isotropic flat plates, or
(ii) the global deformation of the entire web panel modelled as an orthotropic plate. Metwally
(1998) investigated the behavior of steel girders with trapezoidal corrugated webs, and proposed a formula for predicting their nominal shear strength. Yamazaki (2001) proposed formulae for the computation of the buckling strength of corrugated webs, based on results from 6 full-scale models of steel bridge girder webs. Driver et al. (2006) tested full-scale corrugated web girders made of HPS 485W steel, assessed the effect of web initial geometric imperfections through measurements of the out-of-plane displacements, and proposed a lower bound design equation that accounts for both local and global buckling of the web in the elastic and inelastic domains. Watanabe et al. (2007) presented test results for the shear capacity of steel girders using four different trapezoidal corrugation shapes. Yi et al. (2008) studied the nature of the interactive shear buckling of corrugated webs, and concluded that the first order interactive shear buckling equation not accounting for material inelasticity provides a good estimation of the shear strength of corrugated steel webs by comparison with 15 tests and finite element analysis (FEA) results. Moon et al. (2009) presented 3 test results, described the shear strength formula previously presented by Yi et al. (2008), and compared the proposed formula and several other formulas with results from 17 tests. Sause & Braxtan (2011) summarized previously developed formulas for predicting the shear strength of trapezoidal corrugated steel webs, along with the corresponding theory, and proposed a novel formula. Nie et al. (2013) conducted an experimental and analytical study to investigate the shear strength of trapezoidal corrugated steel webs. The experimental program involved testing eight H-shaped steel girders with trapezoidal webs. Hassanein and Kharoob (2013a, b) carried out geometrically and materially nonlinear imperfect analyses (GMNIA) of full-scale bridge girders with corrugated
steel webs failing by shear, and found that when the ratio of flange's thickness to corrugated web's thickness was greater than 3, the boundary conditions between flange and web were approximately fixed; a formula for computing the interactive shear buckling of corrugated webs under fixed boundary conditions was proposed. Leblouba et al. (2017a, b) conducted laboratory tests on a series of corrugated steel web beams to investigate their shear behaviors; three typical failure modes were observed and the failure mechanisms on the post-buckling phase were assessed; besides, five analytical models for the estimation of the critical shear buckling stress based on FE analysis results were proposed and validated against test data.

Despite all the research done on the shear behavior of steel members with corrugated webs, design codes or guidelines, with the Eurocode (2005) and JSCE (1998) as exceptions, are still lacking specifications for this type of structural elements (Papangelis et al. 2017). Moreover, there are still many uncertainties and discrepancies associated with test data and proposed models due to many factors, including geometric imperfections of the web, material properties, shear bucking modes, and inconsistencies between the assumed test conditions and the theoretical models. In order to effectively (accurately and efficiently) estimate the shear capacity of corrugated web steel girders, this paper proposes the use of artificial neural networks (also referred in this manuscript as ANN or neural nets), a popular machine learning technique. Machine learning, one of the six disciplines of Artificial Intelligence (AI) without which the task of having machines acting humanly could not be accomplished, allows us to 'teach' computers how to perform tasks by providing examples of how they should be done (Hertzmann and Fleet 2012). When there is abundant data (also called examples or patterns)
explaining a certain phenomenon, but its theory richness is poor, machine learning can be a
perfect tool; as such, its application to the problem of shear in corrugated steel web is suitable
and timely. The artificial neural network is the (i) oldest (McCulloch and Pitts 1943) and (ii)
most powerful (Hern 2016) technique of machine learning. ANNs also lead the number of
practical applications, virtually covering any field of knowledge (Wilamowski and Irwin 2011,
Prieto et al. 2016). In its most general form, an ANN is a mathematical model designed to
perform a particular task, based in the way the human brain processes information, i.e. with the
help of its processing units (the neurons). ANNs have been employed to perform several types
of real-world basic tasks. Concerning functional approximation, ANN-based solutions are
frequently more accurate than those provided by traditional approaches, such as multi-variate
nonlinear regression, besides not requiring a good knowledge of the function shape being
modelled (Flood 2008). The proposed ANN was designed based on 210 experimental results
available to date in the literature (see section 2). The focus of this study was not to understand
the mechanics underlying the shear behavior of corrugated steel webs, but parametric studies
by means of accurate and robust ANN-based models make it possible to evaluate and improve
existing mechanical models.

## 2. Data Gathering

Many shear strength tests of I-shaped beams and girders with trapezoidal corrugated webs
have been conducted. The 210-point dataset (available in Developer 2018a) used to feed the
ANN software employed in this work, was assembled from the following experimental results:
Lindner & Aschinge (1988), Elgaaly et al. (1996), Peil (1998), Yamazaki (2001), Sause et al.
(2003), Abbas (2003), Sause and Clarke (2003), Gil et al. (2005), Watanabe et al. (2007), Moon
et al. (2009), Sause & Braxtan (2011), He (2011), Nie et al. (2013), Leblouba et al. (2017 a,b).

Seven independent variables were adopted as inputs in ANN simulations, as described and illustrated in Table 1 and Fig. 1, respectively. In Fig.1(a), a simply supported girder with corrugated steel web (height $h_w$) is subjected to a concentrated load (Q) distanced $a$ (shear span) from the left support, until (web) shear failure occurs. Fig.1(b) depicts details of the corrugated web mid surface (top-view of a single “wavelength”), namely the widths of parallel and inclined folds (b and c, respectively), the projected width of the inclined fold (d), the corrugation depth ($h_r$), and the thickness of the corrugated plate ($t_w$). The maximum shear stress (assumed uniform over web’s height) when failure occurs ($\tau_e$) is the target/output (dependent) variable considered in all assessed ANNs. Target values were either reported by the authors of the experimental tests or calculated as $\tau_e = V_e / (h_w t_w)$, being $V_e$ the maximum shear force carried by the test specimen.

<table><caption>**Tab. 1.** Variables (and some stats on their values) considered for ANN simulations.</caption><thead><tr><th rowspan="2">Geometry</th><th rowspan="2">Input variables</th><th rowspan="2">ANN inputs</th><th colspan="3">Values</th></tr><tr><th>min</th><th>max</th><th>average</th></tr></thead><tbody><tr><td rowspan="6"></td><td>a (mm)</td><td>shear span</td><td>1</td><td>287</td><td>4500</td><td>990</td></tr><tr><td>h<sub>w</sub> (mm)</td><td>web height</td><td>2</td><td>260</td><td>2005</td><td>744</td></tr><tr><td>t<sub>w</sub> (mm)</td><td>web thickness</td><td>3</td><td>0.1</td><td>8</td><td>1.6</td></tr><tr><td>b (mm)</td><td>width of the parallel fold</td><td>4</td><td>19.8</td><td>300</td><td>112</td></tr><tr><td>d (mm)</td><td>projected width of the inclined fold</td><td>5</td><td>11.9</td><td>220</td><td>71.8</td></tr><tr><td>h<sub>r</sub> (mm)</td><td>corrugation depth</td><td>6</td><td>12</td><td>150.2</td><td>66.9</td></tr></tbody></table>
Abambres M, He J (2018). Neural network-based analytical model to predict
the shear strength of steel girders with a trapezoidal corrugated web, [hal-02074778](https://hal-02074778)

© 2018 by Abambres M, He J (CC BY 4.0)

<table><tr><td>Material</td><td>f<sub>y</sub> (MPa)</td><td>steel nominal yield stress</td><td>7</td><td>171</td><td>714</td><td>377.69</td></tr><tr><td colspan="3">Target variable</td><td>ANN output</td><td colspan="3">Values</td></tr><tr><td>Strength</td><td>τ<sub>e</sub> (MPa)</td><td>ultimate shear stress</td><td>1</td><td>min</td><td>max</td><td>average</td></tr><tr><td colspan="3"></td><td></td><td>19.0</td><td>375.9</td><td>160.1</td></tr></table>

<figure><img src="image_2.png" /><figcaption>**Fig. 1.** (a) Simple supported corrugated web steel girder under concentrated load, and (b) mid surface profile of a single “wavelength” of the trapezoidal corrugated web.</figcaption></figure>
# 3. Artificial Neural Networks

## 3.1 Introduction

The general ANN structure consists of several nodes disposed in *L* vertical layers (input layer, hidden layers, and output layer) and connected between them, as depicted in Fig. 2. Associated to each node in layers 2 to *L*, also called neuron, is a linear or nonlinear transfer (also called activation) function, which receives the so-called net input and transmits an output. All ANNs implemented in this work are called feedforward, since data presented in the input layer flows in the forward direction only, i.e. every node only connects to nodes belonging to layers located at the right-hand-side of its layer, as shown in Fig. 2. ANN's computing power makes them suitable to efficiently solve small to large-scale complex problems, which can be attributed to their (i) massively parallel distributed structure and (ii) ability to learn and generalize, i.e, produce reasonably accurate outputs for inputs not used during the learning (also called training) phase.

<figure><img src="image_3.png" /><figcaption>**Fig. 2.** Example of a feedforward neural network.</figcaption></figure>
## 3.2 Learning

Each connection between 2 nodes is associated to a synaptic weight (real value), which, together with each neuron's bias (also a real value), are the most common types of neural net unknown parameters that will be determined through learning. Learning is nothing else than determining network unknown parameters through some algorithm in order to minimize network's performance measure, typically a function of the difference between predicted and target (desired) outputs. When ANN learning has an iterative nature, it consists of three phases: (i) training, (ii) validation, and (iii) testing. From previous knowledge, examples or data points are selected to train the neural net, grouped in the so-called training dataset. Those examples are said to be 'labelled' or 'unlabeled', whether they consist of inputs paired with their targets, or just of the inputs themselves – learning is called supervised (e.g., functional approximation, classification) or unsupervised (e.g., clustering), whether data used is labelled or unlabeled, respectively. During an iterative learning, while the training dataset is used to tune network unknowns, a process of cross-validation takes place by using a set of data completely distinct from the training counterpart (the validation dataset), so that the generalization performance of the network can be attested. Once 'optimum' network parameters are determined, typically associated to a minimum of the validation performance curve (called early stop – see Fig. 3), many authors still perform a final assessment of model's accuracy, by presenting to it a third fully distinct dataset called 'testing'. Heuristics suggests that early stopping avoids overfitting, i.e. the loss of ANN's generalization ability. One of the causes of overfitting might be learning
too many input-target examples suffering from data noise, since the network might learn some
of its features, which do not belong to the underlying function being modelled (Haykin 2009).

<figure><img src="image_4.png" /><figcaption>**Fig. 3.** Cross-validation - assessing network's generalization ability.</figcaption></figure>

**3.3 Implemented ANN features**

The ‘behavior’ of any ANN depends on many ‘features’, having been implemented 15 ANN
features in this work (including data pre/post processing ones). For those features, it is important
to bear in mind that no ANN guarantees good approximations via extrapolation (either in functional
approximation or classification problems), i.e. the implemented ANNs should not be applied
outside the input variable ranges used for network training. Since there are no objective rules
dictating which method per feature guarantees the best network performance for a specific
problem, an extensive parametric analysis (composed of nine parametric sub-analyses) was carried
out to find ‘the optimum’ net design. A description of all methods/formulations implemented
for each ANN feature (see Tabs. 2-4) – they are a selection from state of art literature on ANNs,
including both traditional and promising modern techniques, can be found in previous
published works (e.g., Abambres et al. 2018) – the reader might need to go through it to fully
understand the meaning of all variables reported in this manuscript. The whole work was coded in MATLAB (The Mathworks, Inc. 2017), making use of its neural network toolbox when dealing with popular learning algorithms (1-3 in Tab. 4). Each parametric sub-analysis (SA) consists of running all feasible combinations (also called ‘combos’) of pre-selected methods for each ANN feature, in order to get performance results for each designed net, thus allowing the selection of the best ANN according to a certain criterion. The best network in each parametric SA is the one exhibiting the smallest average relative error (called performance) for all learning data.

<table><caption>**Tab. 2.** Implemented ANN features (F) 1-5.</caption><thead><tr><th rowspan="2">FEATURE METHOD</th><th colspan="2">F1</th><th colspan="2">F2</th><th colspan="2">F3</th><th colspan="2">F4</th><th colspan="2">F5</th></tr><tr><th>Qualitative Var Represent</th><th>Dimensional Analysis</th><th>Input Dimensionality Reduction</th><th>% Train-Valid-Test</th><th>Input Dimensionality Reduction</th><th>% Train-Valid-Test</th><th>Input Dimensionality Reduction</th><th>Input Normalization</th><th>Input Dimensionality Reduction</th><th>Input Normalization</th></tr></thead><tbody><tr><td>1</td><td>Boolean Vectors</td><td>Yes</td><td>Linear Correlation</td><td>80-10-10</td><td>Linear Max Abs</td><td>80-10-10</td><td>Linear Max Abs</td><td>80-10-10</td><td>Linear Max Abs</td><td>80-10-10</td></tr><tr><td>2</td><td>Eq Spaced in [0,1]</td><td>No</td><td>Auto-Encoder</td><td>70-15-15</td><td>Linear [0, 1]</td><td>70-15-15</td><td>Linear [0, 1]</td><td>70-15-15</td><td>Linear [0, 1]</td><td>70-15-15</td></tr><tr><td>3</td><td>-</td><td>-</td><td>-</td><td>60-20-20</td><td>Linear [-1, 1]</td><td>60-20-20</td><td>Linear [-1, 1]</td><td>60-20-20</td><td>Linear [-1, 1]</td><td>60-20-20</td></tr><tr><td>4</td><td>-</td><td>-</td><td>Ortho Rand Proj</td><td>50-25-25</td><td>Nonlinear</td><td>50-25-25</td><td>Nonlinear</td><td>50-25-25</td><td>Nonlinear</td><td>50-25-25</td></tr><tr><td>5</td><td>-</td><td>-</td><td>Sparse Rand Proj</td><td>-</td><td>Lin Mean Std</td><td>-</td><td>Lin Mean Std</td><td>-</td><td>Lin Mean Std</td><td>-</td></tr><tr><td>6</td><td>-</td><td>-</td><td>No</td><td>-</td><td>No</td><td>-</td><td>No</td><td>-</td><td>No</td><td>-</td></tr></tbody></table>

<table><caption>**Tab. 3.** Implemented ANN features (F) 6-10.</caption><thead><tr><th rowspan="2">FEATURE METHOD</th><th colspan="2">F6</th><th colspan="2">F7</th><th colspan="2">F8</th><th colspan="2">F9</th><th colspan="2">F10</th></tr><tr><th>Output Transfer</th><th>Output Normalization</th><th>Net Architecture</th><th>Hidden Layers</th><th>Connectivity</th><th>MLPN</th><th>1 HL</th><th>2 HL</th><th>Adj Layers + In-Out</th><th>Fully-Connected</th></tr></thead><tbody><tr><td>1</td><td>Logistic</td><td>Lin [a, b] = 0.7[φ<sub>min</sub>, φ<sub>max</sub>]</td><td>RBFN</td><td>3 HL</td><td>Adj Layers + In-Out</td><td>MLPN</td><td>1 HL</td><td>2 HL</td><td>Adj Layers + In-Out</td><td>Fully-Connected</td></tr><tr><td>2</td><td>-</td><td>Lin [a, b] = 0.6[φ<sub>min</sub>, φ<sub>max</sub>]</td><td>RBFN</td><td>3 HL</td><td>Adj Layers + In-Out</td><td>RBFN</td><td>1 HL</td><td>2 HL</td><td>Adj Layers + In-Out</td><td>Fully-Connected</td></tr><tr><td>3</td><td>Hyperbolic Tang</td><td>Lin [a, b] = 0.5[φ<sub>min</sub>, φ<sub>max</sub>]</td><td>-</td><td>3 HL</td><td>Fully-Connected</td><td>-</td><td>1 HL</td><td>2 HL</td><td>Adj Layers + In-Out</td><td>Fully-Connected</td></tr><tr><td>4</td><td>-</td><td>Linear Mean Std</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>5</td><td>Bilinear</td><td>No</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>6</td><td>Compet</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>7</td><td>Identity</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr></tbody></table>
Abambres M, He J (2018). Neural network-based analytical model to predict the shear strength of steel girders with a trapezoidal corrugated web, hal-02074778
© 2018 by Abambres M, He J (CC BY 4.0)

<table><caption>Tab. 4. Implemented ANN features (F) 11-15.</caption><thead><tr><th rowspan="2">FEATURE METHOD</th><th colspan="2">F11</th><th colspan="3">F12</th><th colspan="2">F13</th><th colspan="2">F14</th><th colspan="2">F15</th></tr><tr><th>Hidden Transfer</th><th>Parameter Initialization</th><th>Learning Algorithm</th><th>Performance Improvement</th><th>Training Mode</th><th>NNC</th><th>Batch</th><th>Mini-Batch</th><th>Online</th><th>Batch</th><th>Mini-Batch</th></tr></thead><tbody><tr><td>1</td><td>Logistic</td><td>Midpoint (W) + Rands (b)</td><td>BP</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>2</td><td>Identity-Logistic</td><td>Rands</td><td>BPA</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>3</td><td>Hyperbolic Tang</td><td>Randnc (W) + Rands (b)</td><td>LM</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>4</td><td>Bipolar</td><td>Randnr (W) + Rands (b)</td><td>ELM</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>5</td><td>Bilinear</td><td>Randsmall</td><td>mb ELM</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>6</td><td>Positive Sat Linear</td><td>Rand [-Δ, Δ]</td><td>I-ELM</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>7</td><td>Sinusoid</td><td>SVD</td><td>CI-ELM</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>8</td><td>Thin-Plate Spline</td><td>MB SVD</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>9</td><td>Gaussian</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>10</td><td>Multiquadratic</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>11</td><td>Radbas</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr></tbody></table>

## 3.4 Network Performance Assessment

Several types of results were computed to assess network outputs, namely (i) maximum error, (ii) % errors greater than 3%, and (iii) performance, which are defined next. All abovementioned errors are relative errors (expressed in %) based on the following definition, concerning a single output variable and data pattern,

$$e_{qp} = 100 \left| \frac{d_{qp} - y_{qLp}}{d_{qp}} \right|, \quad (1)$$

where (i) $d_{qp}$ is the $q^{th}$ desired (or target) output when pattern $p$ within iteration $i$ ($p=1,..., P_i$) is presented to the network, and (ii) $y_{qLp}$ is net's $q^{th}$ output for the same data pattern. Moreover, denominator in eq. (1) is replaced by 1 whenever $|d_{qp}| < 0.05 - d_{qp}$ in the nominator keeps its
real value. This exception to eq. (1) aims to reduce the apparent negative effect of large relative
errors associated to target values close to zero. Even so, this trick may still lead to (relatively)
large solution errors while groundbreaking results are depicted as regression plots (target vs.
predicted outputs).

**3.4.1 Maximum Error**

This variable measures the maximum relative error, as defined by eq. (1), among all output
variables and learning patterns.

**3.4.2 Percentage of Errors > 3%**

This variable measures the percentage of relative errors, as defined by eq. (1), among all
output variables and learning patterns, that are greater than 3%.

**3.4.3 Performance**

In functional approximation problems, network performance is defined as the average
relative error, as defined in eq. (1), among all output variables and data patterns being evaluated
(e.g., training, all data).

**3.5 Software Validation**

Several benchmark datasets/functions were used to validate the developed software, involving
low- to high-dimensional problems and small to large volumes of data. Due to paper length limit,
validation results are not presented herein but they were made public online (Researcher 2018b).
## 3.6 Parametric Analysis Results

Aiming to reduce the computing time by cutting in the number of combos to be run – note that all features combined lead to hundreds of millions of combos, the whole parametric simulation was divided into nine parametric SAs, where in each one feature 7 only takes a single value. This measure aims to make the performance ranking of all combos within each ‘small’ analysis more ‘reliable’, since results used for comparison are based on target and output datasets as used in ANN training and yielded by the designed network, respectively (they are free of any postprocessing that eliminates output normalization effects on relative error values). Whereas (i) the 1st and 2nd SAs aimed to select the best methods from features 1, 2, 5, 8 and 13 (all combined), while adopting a single popular method for each of the remaining features (F<sub>3</sub>: 6, F<sub>4</sub>: 2, F<sub>6</sub>: {1 or 7}, F<sub>7</sub>: 1, F<sub>9</sub>: 1, F<sub>10</sub>: 1, F<sub>11</sub>: {3, 9 or 11}, F<sub>12</sub>: 2, F<sub>14</sub>: 1, F<sub>15</sub>: 1 – see Tabs. 2-4) – SA 1 involved learning algorithms 1-3 and SA 2 involved the ELM-based counterpart, (ii) the 3rd – 7th SAs combined all possible methods from features 3, 4, 6 and 7, and concerning all other features, adopted the methods integrating the best combination from the aforementioned first SA, (iii) the 8th SA combined all possible methods from features 11, 12 and 14, and concerning all other features, adopted the methods integrating the best combination (results compared after postprocessing) among the previous five sub-analyses, and lastly (iv) the 9th SA combined all possible methods from features 9, 10 and 15, and concerning all other features, adopted the methods integrating the best combination from the previous analysis. Summing up the ANN feature combinations for all parametric SAs, a total of 475 combos were run for this work.
ANN feature methods used in the best combo from each of the abovementioned nine parametric sub-analyses, are specified in Tab. 5 (the numbers represent the method number as in Tabs 2-4). Tab. 6 shows the corresponding relevant results for those combos, namely (i) maximum error, (ii) % errors > 3%, (iii) performance (all described in section 3, and evaluated for all learning data), (iv) total number of hidden nodes in the model, and (v) average computing time per example (including data pre- and post-processing). All results shown in Tab. 6 (i) were obtained for single ANNs only, since no NNC networks yielded better results for this particular problem, and (ii) are based on target and output datasets computed in their original format, i.e. free of any transformations due to output normalization and/or dimensional analysis. The microprocessors used in this work have the following features: OS: Win10Home 64bits, RAMs: 48 GB, Local Disk Memory: 1 TB, CPUs: Intel® Core™ i7 8700K @ 3.70-4.70 GHz.

<table><caption>**Tab. 5.** ANN feature (F) methods used in the best combo from each parametric sub-analysis (SA).</caption><thead><tr><th>SA</th><th>F1</th><th>F2</th><th>F3</th><th>F4</th><th>F5</th><th>F6</th><th>F7</th><th>F8</th><th>F9</th><th>F10</th><th>F11</th><th>F12</th><th>F13</th><th>F14</th><th>F15</th></tr></thead><tbody><tr><td>1</td><td>1</td><td>2</td><td>6</td><td>2</td><td>5</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>3</td><td>2</td><td>3</td><td>1</td><td>3</td></tr><tr><td>2</td><td>1</td><td>2</td><td>6</td><td>2</td><td>2</td><td>7</td><td>1</td><td>2</td><td>1</td><td>1</td><td>9</td><td>2</td><td>5</td><td>1</td><td>3</td></tr><tr><td>3</td><td>1</td><td>2</td><td>6</td><td>3</td><td>5</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>3</td><td>2</td><td>3</td><td>1</td><td>3</td></tr><tr><td>4</td><td>1</td><td>2</td><td>1</td><td>1</td><td>5</td><td>1</td><td>2</td><td>1</td><td>1</td><td>1</td><td>3</td><td>2</td><td>3</td><td>1</td><td>3</td></tr><tr><td>5</td><td>1</td><td>2</td><td>1</td><td>2</td><td>5</td><td>1</td><td>3</td><td>1</td><td>1</td><td>1</td><td>3</td><td>2</td><td>3</td><td>1</td><td>3</td></tr><tr><td>6</td><td>1</td><td>2</td><td>6</td><td>4</td><td>5</td><td>7</td><td>4</td><td>1</td><td>1</td><td>1</td><td>3</td><td>2</td><td>3</td><td>1</td><td>3</td></tr><tr><td>7</td><td>1</td><td>2</td><td>6</td><td>3</td><td>5</td><td>5</td><td>5</td><td>1</td><td>1</td><td>1</td><td>3</td><td>2</td><td>3</td><td>1</td><td>3</td></tr><tr><td>8</td><td>1</td><td>2</td><td>6</td><td>3</td><td>5</td><td>5</td><td>5</td><td>1</td><td>1</td><td>1</td><td>3</td><td>5</td><td>3</td><td>1</td><td>3</td></tr><tr><td>9</td><td>1</td><td>2</td><td>6</td><td>3</td><td>5</td><td>5</td><td>5</td><td>1</td><td>2</td><td>3</td><td>3</td><td>5</td><td>3</td><td>1</td><td>3</td></tr></tbody></table>
<table><caption>**Tab. 6.** Performance results for the best design from each parametric sub-analysis.</caption><thead><tr><th rowspan="2">SA</th><th colspan="4">ANN</th><th rowspan="2">Running Time / Data Point (s)</th></tr><tr><th>Max Error (%)</th><th>Performance All Data (%)</th><th>Errors > 3% (%)</th><th>Total Hidden Nodes</th></tr></thead><tbody><tr><td>1</td><td>0.7</td><td>0.0</td><td>0.0</td><td>28</td><td>1.93E-04</td></tr><tr><td>2</td><td>265.3</td><td>15.6</td><td>72.4</td><td>90</td><td>9.80E-05</td></tr><tr><td>3</td><td>0.8</td><td>0.0</td><td>0.0</td><td>28</td><td>1.51E-04</td></tr><tr><td>4</td><td>0.9</td><td>0.0</td><td>0.0</td><td>28</td><td>1.29E-04</td></tr><tr><td>5</td><td>0.8</td><td>0.0</td><td>0.0</td><td>28</td><td>1.44E-04</td></tr><tr><td>6</td><td>0.1</td><td>0.0</td><td>0.0</td><td>28</td><td>1.01E-04</td></tr><tr><td>7</td><td>26.6</td><td>1.5</td><td>14.3</td><td>28</td><td>1.26E-04</td></tr><tr><td>8</td><td>17.0</td><td>1.2</td><td>12.4</td><td>28</td><td>1.41E-04</td></tr><tr><td>9</td><td>0.0</td><td>0.0</td><td>0.0</td><td>28</td><td>1.35E-04</td></tr></tbody></table>

## 3.7 Proposed ANN-Based Model

The proposed model is the one, among the best ones from all parametric SAs, exhibiting the lowest maximum error (SA 9). That model is characterized by the ANN feature methods {1, 2, 6, 3, 5, 5, 5, 1, 2, 3, 3, 5, 3, 1, 3} in Tabs. 2-4. Aiming to allow implementation of this model by any user, all variables/equations required for (i) data preprocessing, (ii) ANN simulation, and (iii) data postprocessing, are presented in 3.7.1-3.7.3, respectively. The proposed model is a single MLPN with 4 layers and a distribution of nodes/layer of 7-14-14-1. Concerning connectivity, the network is fully-connected, and the hidden and output transfer functions are all Hyperbolic Tangent and Bilinear, respectively. The network was trained using the Levenberg-Marquardt algorithm (978 epochs). After design, the average network computing time concerning the presentation of a single example (including data pre/postprocessing) is 1.35E-04 s – Fig. 4 depicts a simplified scheme of some of network key features. Lastly, all relevant performance results concerning the proposed
ANN are illustrated in 3.7.4. The obtained ANN solution for every data point can be found in
Developer (2018a).

<figure><img src="image_5.png" /><figcaption>**Fig. 4** Proposed 7-14-14-1 fully-connected MLPN – simplified scheme.</figcaption></figure>

It is worth recalling that, in this manuscript, whenever a vector is added to a matrix, it means
the former is to be added to all columns of the latter (valid in MATLAB).

**3.7.1 Input Data Preprocessing**

For future use of the proposed ANN to simulate new data $Y_{1,sim}$ (7 x $P_{sim}$ matrix), concerning $P_{sim}$ patterns, the same data preprocessing (if any) performed before training must be applied to the input dataset. That preprocessing is defined by the methods used for ANN features 2, 3 and 5 (respectively 2, 6 and 5 – see Tab. 2), which should be applied after all (eventual) qualitative variables in the input dataset are converted to numerical (using feature 1's method). Next, the necessary preprocessing to be applied to $Y_{1,sim}$, concerning features 2, 3 and 5, is fully described.
## Dimensional Analysis and Dimensionality Reduction

Since dimensional analysis (d.a.) and dimensionality reduction (d.r.) were not carried out,
one has

$$ \{Y_{1,sim}\}_{d.r.}^{after} = \{Y_{1,sim}\}_{d.a.}^{after} = Y_{1,sim} \quad . \quad (2) $$

## Input Normalization

After input normalization, the new input dataset {$Y_{1,sim}$}$_n$ after is defined as function of the previously determined {$Y_{1,sim}$}$_{d.r}$ after, and they have the same size, reading

$$ \{Y_{1,sim}\}_n^{after} = \left( \{Y_{1,sim}\}_{d.r.}^{after} - \text{INP}(:,1) \right) / \text{INP}(:,2) \\ \text{INP} = \begin{bmatrix} 990.400346190476 & 668.905673826269 \\ 744.012857142857 & 396.481809107959 \\ 1.55576190476190 & 1.28849686190348 \\ 111.9866666666667 & 71.5898412598522 \\ 71.8395776190476 & 52.3896551822015 \\ 66.8513071428572 & 47.7496285409415 \\ 377.6900000000000 & 133.759804523706 \end{bmatrix} \quad , \quad (3) $$

where one recalls that operator ‘./’ divides row i in the numerator by INP(i, 2).

## 3.7.2 ANN-Based Analytical Model

Once determined the preprocessed input dataset {$Y_{1,sim}$}$_n$ after (7 x $P_{sim}$ matrix), the next step is to present it to the proposed ANN to obtain the predicted output dataset {$Y_{4,sim}$}$_n$ after (1 x $P_{sim}$
vector), which will be given in the same preprocessed format of the target dataset used in learning. In order to convert the predicted outputs to their ‘original format’ (i.e., without any transformation due to normalization or dimensional analysis – the only transformation visible will be the (eventual) qualitative variables written in their numeric representation), some postprocessing is needed, as described in detail in 3.7.3. Next, the mathematical representation of the proposed ANN is given, so that any user can implement it to determine {$Y_{4,sim}$$^{after}$, thus eliminating all rumors that ANNs are ‘black boxes’.

$$
\begin{align}
Y_2 &= \varphi_2 \left( W_{1-2}^T \left\{ Y_{1,sim} \right\}_n^{after} + b_2 \right) \nonumber \\
Y_3 &= \varphi_3 \left( W_{1-3}^T \left\{ Y_{1,sim} \right\}_n^{after} + W_{2-3}^T Y_2 + b_3 \right) \nonumber \\
\left\{ Y_{4,sim} \right\}_n^{after} &= \varphi_4 \left( W_{1-4}^T \left\{ Y_{1,sim} \right\}_n^{after} + W_{2-4}^T Y_2 + W_{3-4}^T Y_3 + b_4 \right) \tag{4}
\end{align}
$$

where

$$
\varphi_2 = \varphi_3 = \varphi(s) = \frac{e^s - e^{-s}}{e^s + e^{-s}}
$$
$$
\varphi_4 = \varphi_4(s) = \begin{cases} s, & s \ge 0 \\ 0, & s < 0 \end{cases} \qquad . (5)
$$

Arrays $W_{j-s}$ and $b_s$ are stored online in Developer (2018b), aiming to avoid an overlong article and ease model's implementation by any interested reader.
### 3.7.3 Output Data Postprocessing

In order to transform the output dataset obtained by the proposed ANN, {$Y_{4,sim}$}$_n$ after (1 x $P_{sim}$ vector), to its original format ($Y_{4,sim}$), i.e. without the effects of dimensional analysis and/or output normalization (possibly) taken in target dataset preprocessing prior training, one has

$$Y_{4,sim} = \{Y_{4,sim}\}_{n}^{\text{after}} \quad , \quad (6)$$

since no output normalization nor dimensional analysis were carried out.

### 3.7.4 Performance Results

Finally, results yielded by the proposed ANN, in terms of performance variables defined in sub-section 3.4, are presented in this section in the form of several graphs: (i) a regression plot (Fig. 5) where network target and output data are plotted, for each data point, as x- and y-coordinates respectively – a measure of linear correlation is given by the Pearson Correlation Coefficient (R); (ii) a performance plot (Fig. 6), where performance (average error) values are displayed for several learning datasets; and (iii) an error plot (Fig. 7), where values concern all data (iii₁) maximum error and (iii₂) % of errors greater than 3%.
Abambres M, He J (2018). Neural network-based analytical model to predict
the shear strength of steel girders with a trapezoidal corrugated web, [hal-02074778](https://hal-02074778)

© 2018 by Abambres M, He J (CC BY 4.0)

<figure><img src="image_6.png" /><figcaption>**Fig. 5.** Regression plot for the proposed ANN (see output variable in Fig. 1).</figcaption></figure>
Abambres M, He J (2018). Neural network-based analytical model to predict
the shear strength of steel girders with a trapezoidal corrugated web, [hal-02074778](https://hal-02074778)

© 2018 by Abambres M, He J (CC BY 4.0)

<figure><img src="image_7.png" /><figcaption>**Fig. 6.** Performance plot (mean errors) for the proposed ANN.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Fig. 7. Error plot for the proposed ANN.</figcaption></figure>

**4. ANN-based vs. Existing Models**

Shear strength of steel I-girders is controlled by buckling and/or shear yielding of the corrugated web. Shear buckling of corrugated webs is often classified as local buckling, global buckling, and interactive buckling, as exemplified in Fig.8 via the FEA-based displacement contours for those elastic buckling modes. Global buckling involves multiple folds and the buckled shape extends diagonally over the height of the web. Local buckling is controlled by deformations within a single “fold” of the web. The interactive shear buckling mode is attributed to the interaction between local and global shear buckling modes.
Abambres M, He J (2018). Neural network-based analytical model to predict
the shear strength of steel girders with a trapezoidal corrugated web, [hal-02074778](https://hal-02074778)

© 2018 by Abambres M, He J (CC BY 4.0)

<figure><img src="image_9.png" /><figcaption>**Fig.8.** Shear buckling modes via FEA.</figcaption></figure>

**4.1 Local shear buckling**

The local elastic shear buckling stress of a corrugated web can be predicted using classic
plate buckling theory (Timoshenko and Gere 1961). A single parallel or inclined fold is assumed to be supported by the adjacent folds and steel flanges. The corresponding local elastic shear buckling stress, $\tau_{cr,L}^e$ is

$$ \tau_{cr,L}^e = k_L \frac{\pi^2 E}{12(1-\nu^2)} \left(\frac{t_w}{w}\right)^2, \quad (7) $$

where (i) $w$ is the maximum fold width, $\max(b, c)$, and (ii) $k_L$ is the local shear buckling coefficient, which depends on the boundary conditions and the fold aspect ratio – $k_L$ lies between 5.34 (assuming simply supported edges) and 8.98 (assuming fixed edges). For practical design purposes, $k_L=5.34$ is recommend by Moon et al. (2009).

## 4.2 Global shear buckling

An expression for the global elastic shear buckling stress of a corrugated steel plate ($\tau_{cr,G}^e$) was developed by Easley and McFarland (1969) using orthotropic plate theory, reading

$$ \tau_{cr,G}^e = k_G \frac{D_x^{3/4} D_y^{1/4}}{t_w h_w^2}, \quad (8) $$

being (i) $k_G$, the global shear buckling coefficient, and (ii) $D_x$ and $D_y$, the bending stiffnesses per unit length of the corrugated web with respect to its central principal axes x and y, respectively. Easley (1975) proposed that $k_G$ varies between 36 (assuming the web pin-ended by the flanges) and 68.4 (assuming the web fix-ended by the flanges). Elgaaly et al (1996)
suggested that $k_G$ should be taken as 31.6 for simply supported boundaries and 59.2 for the clamped counterpart. $D_x$ and $D_y$ for trapezoidal corrugated webs can be determined as:

$$D_x = \frac{EI_x}{b+h_r} \qquad (9)$$

$$D_y = \frac{Et_w^3 b+d}{12 b+c} \qquad (10)$$

where $I_x$ is the moment of inertia about the x-axis, $I_x = 2bt_w(h_r/2)^2 + h_r^3 t_w / (6\sin\alpha)$.

## 4.3 Interactive shear buckling

The interactive shear buckling mode is attributed to the interaction between local and global modes and governs shear buckling strength. Lindner & Aschinger (1988) first proposed the corresponding elastic shear buckling stress formula ($\tau_{cr,I}^e$)

$$\tau_{cr,I}^e = \frac{\tau_{cr,L}^e \cdot \tau_{cr,G}^e}{\left( \left( \tau_{cr,G}^e \right)^n + \left( \tau_{cr,L}^e \right)^n \right)^{1/n}} \qquad (11)$$

being exponent $n$ an integer – several researchers have proposed distinct values (Yi et al. 2008, Sause & Braxtan 2011, Hassanein & Kharoob 2013a, Leblouba et al. 2017b).

## 4.4 Shear strength

Previous studies (Yi et al. 2008, Sause & Braxtan 2011) have shown that shear strength of corrugated steel webs was generally controlled by interactive shear buckling. In this context,
the shear buckling non-dimensional slenderness ($\lambda_{I,n}$) of a corrugated steel web is defined as

$$ \lambda_I = \sqrt{\tau_y / \tau_{cr,I}^e} \quad , \quad (12) $$

where $\tau_y$ is the tangential yield stress (typically $f_y / \sqrt{3}$, being $f_y$ the normal yield stress).

Driver et al. (2006) proposed eq. (11) to calculate the shear capacity of corrugated steel webs for all types of shear failures, reading (n=2)

$$ \tau_D = \sqrt{\frac{(\tau_{cr,L}^e \cdot \tau_{cr,G}^e)^2}{(\tau_{cr,G}^e)^2 + (\tau_{cr,L}^e)^2}} \quad . \quad (13) $$

In case any elastic shear buckling stress ($\tau_{cr,G}^e$ or $\tau_{cr,L}^e$) exceeds 80% of the shear yield stress $\tau_y$, its value in eq. (13) should be replaced by an inelastic counterpart given by (Elgaaly et al. 1996)

$$ \tau_{cr}^{in} = \sqrt{0.8 \tau_y \tau_{cr}^e} \le \tau_y \quad . \quad (14) $$

El-Metwally (1998) proposed the following equation to calculate the shear capacity of corrugated steel webs (using n=2 in eq. (11))

$$ \tau_{EL} = \tau_y \times \left( 1 / \left( (\lambda_I)^4 + 1 \right) \right)^{1/2} \quad . \quad (15) $$

Sauce and Braxtan (2011) summarized a large number of previous experimental data, and selected 22 groups of results to fit eq. (16), for the prediction of the shear capacity of corrugated steel webs (using n=3 in eq. (11))

$$ \tau_{\text{Sauce}} = \tau_y \times \left( 1 / \left( (\lambda_I)^6 + 2 \right) \right)^{1/3} \quad . \quad (16) $$

Leblouba et al. (2017a), based on 113 test results collected from the literature, and 12 tests
carried out by themselves, developed the following analytical model (based on the hyperbolic Richards equation) to calculate the shear strength of corrugated steel webs (using n=4 in eq. (11))

$$ \tau_{\text{Leblouba}} = \tau_y \times \left( 1 / \left( \left( \lambda_l / 1.58 \right)^{1.6} + 1 \right) \right)^{1.15} \quad (17) $$

Fig.9 compares the shear strengths yielded by the analytical models presented before ($\tau_D$, $\tau_{EL}$, $\tau_{Sauce}$, $\tau_{Leblouba}$, $\tau_{ANN}$) to those obtained experimentally ($\tau_e$), for the 210 steel girders assessed in this work (dataset in Developer 2018a). The average ratios $\tau_D/\tau_e$, $\tau_{EL}/\tau_e$, $\tau_{Sauce}/\tau_e$, $\tau_{Leblouba}/\tau_e$ are 0.78, 0.87, 0.82 and 0.83, with standard deviations of 0.13, 0.16, 0.13 and 0.11, respectively. It can be found that all those models underestimate the shear strength of trapezoidal corrugated web girders. For comparison, the average value of $\tau_{ANN}/\tau_e$ is 1.00, with a standard deviation of 0.00. The major improvement of the proposed ANN-based analytical model (see sub-section 3.7), as compared to the existing calculation methods, is quite clear in Fig. 9, where x-axis shows the predicted shear capacity $\tau_{model}$ ($\tau_{ANN}$, $\tau_D$, $\tau_{EL}$, $\tau_{Sauce}$, $\tau_{Leblouba}$) and the y-axis shows the experimental counterpart $\tau_e$.

<figure><img src="image_10.png" /></figure>
<figure><img src="image_11.png" /><figcaption>**Fig. 9.** Comparison between tested and predicted shear strength for several proposed models.</figcaption></figure>

**5. Discussion**

Regardless the high quality of the predictions yielded by the proposed model, the reader
should not **blindly** accept it as accurate for any other instances falling inside the input domain
of the design dataset. Any analytical approximation model must undergo extensive validation
before it can be taken as reliable (the more inputs, the larger the validation process). Models
proposed meanwhile are part of a learning process towards excellence.

**6. Conclusions**

This paper describes how artificial neural networks (ANN) can be used to predict the shear
capacity of steel girders with a trapezoidal corrugated web, and proposes an analytical model
for that purpose. The developed model was designed from a 210-point database of test results
available in the literature. Seven governing (geometrical and material) parameters were
identified as input variables, and the ultimate shear stress (assumed uniform along web's
height) at failure was considered as the target/output variable for the ANN simulations. The proposed ANN-based analytical model yielded maximum and mean relative errors of 0.0% concerning all the 210 test results previously collected. Fig. 9 shows that the ANN-based approach clearly outperforms the existing calculation models assessed in this work, for the dataset considered (made available at Developer 2018a) – latter models exhibit mean errors greater than 13%.

The focus of this study was not to assess the mechanics underlying the behavior of corrugated web steel girders, but parametric studies by means of accurate and robust ANN-based models make it possible to evaluate and improve existing mechanical models.

## Contributions

He J. developed sections 1, 2 and 4; Abambres M. developed sections 3 and 5 (ANN-related); Remaining sections had equal contributions from both authors.

## Acknowledgements

There are no conflicts of interest to disclose.

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© 2018 by Abambres M, He J (CC BY 4.0)

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