Wind-structure interaction data-driven neural-network model proposition for assessing performance of unique stiffening-helix against wind-induced open-top tank-buckling

Abstract

Empty steel-tanks are very much susceptible against buckling induced by wind loading. Ring and vertical stiffeners are commonly used to provide necessary strength to thin-walled steel tanks to resist wind-induced buckling. The authors have studied the performance of a unique, ribonucleic acid structure-inspired, stiffening-helix mechanism against the wind-induced buckling of open-top, cylindrical, empty, steel-tank. The most important output parameter of this study is the load multiplier (λ) of buckling, as it defines the stability of tank-shell against wind-induced buckling. The study variables are tank-height to tank-diameter (H/D) ratio, tank-radius to wall-thickness (r/t) ratio, basic wind speed (V_b) and helix pitch length to tank-height (L_P/H) ratio. This study has been performed through multiphysics system-coupling of computational fluid dynamics and structural mechanics (eigenvalue buckling). The stiffening-helix can provide necessary strength to open-top, cylindrical, steel-tank economically against wind-induced buckling. An artificial neural network (ANN) has been trained with the analytical data to develop a predictive model. The proposed predictive ANN model produces 99.11% average accuracy.

Graphical abstract

Keywords

open-top steel-tank / cylindrical steel-tank / stiffening-helix / wind-induced buckling / buckling load multiplier / artificial neural network

1 Introduction

The atmospheric wind harms an empty, cylindrical steel-tank by transverse wind load-induced bending and over-turning, oscillating wind flow-excited ovalling, and stagnation pressure-caused membrane force-induced buckling [1–5]. To protect the tank from overturning base of the tank is anchored by bolts or straps [6,7]. To minimize the bending and axial buckling of tank, vertical stiffener is useful and its performance can be improved with the application of corrugated shell course at the tank-wall [6,8]. Ring stiffener is applied to control the ovalling and circumferential buckling of tank wall [6,9]. Determination of the permissible height of unstiffened portion of shell and equivalent thickness of stepped wall from varying thickness of shell courses are important to assess the need of intermediate ring-stiffener [10,11]. Through simplified modeling of ring-stiffener through linear bifurcation, material and geometric nonlinear analyses using beam element with translational constraints can increase destabilization risk of tank by 7%, 12%, and 14% against thermal, wind and uniform loading [12]. But such modeling simplification does not alter the critical stress state of tank buckling significantly and can be faithful to the real scenario [13]. Top and intermediate ring stiffeners help in minimizing deformation at the tank’s upper and intermediate portions, respectively [9]. The top stiffener is usually provided with greater size compared to the intermediate stiffener, but later studies highlight that similar size of both stiffeners does not affect the tank’s strength against buckling [14–18]. Modifications in the design formulae of permissible height of unstiffened shell-portion; moment of inertia, section modulus, radial shear, bending moment and stresses in both type of stiffener; internal spacing of intermediate stiffeners [19–22]. The concept of spiral stiffening of cylindrical shell under uniform torsion, compression and transverse loading, comes to light from the study of Yen [23] on the optimization of stiffener weight depending on the spiral angle. 30° angled spiral stiffener improves the response against axial buckling of ceramic shell on elastic Winkler foundation [24]. When spiral stiffener with spiral angle < 60° is attached at the inside face of the wall it reduces tank’s capacity against thermal buckling compared to the externally connected spiral stiffener [25]. Higher resistance to thermal load induced large deflection can be seen in spirally stiffened functionally graded shell with damper, situated on elastic foundation, when the load is linearly varying, rather than uniform [26,27]. Through comparative study, Nam et al. [28] proved that spiral stiffener renders higher strength to shell against buckling, than orthogonal stiffener. Amalgamation of Von Karman’s concept of nonlinearity with Donnel’s shell-buckling theory forms the problem equations of the thermo-torsional buckling of shell with spiral stiffening mechanism based on Lekhnitskii’s smeared stiffening approach, which can be solved with Galerkin method [29,30]. Exploring the applications of vertical, ring and spiral stiffeners it is evident that usage of ring stiffener is much more in practice for resisting wind-induced buckling damage and providing stability to cylindrical steel-tanks. Vertical stiffeners are applicable for resisting axial buckling of steel-tanks and less likely to be used against wind-induced buckling. Application of spiral stiffener is reported for the axial, thermal and torsional buckling of composite shells only and the performance of this type of stiffener is not tested or studied against wind-induced steel-tank buckling. This scope encouraged the authors to study the performance of stiffening-helix against the wind-induced tank buckling, which is inspired from the structure of ribonucleic acid (RNA). The study is performed by varying the tank’s height to diameter (H/D) and radius to wall-thickness (r/t) ratio. The basic speed of wind (V_b) and pitch length of the stiffening-helix to tank height (L_p/H) ratio have also been varied in this study. The performance parameters are normalized radial deformation ({\delta }_{{\mathrm{r}}_{\mathrm{B}\mathrm{N}}}) to assess damage and buckling load multiplier (λ) for understanding stability of tank. This study is not focused on providing designing specification for the stiffening-helix and American petroleum institute (API) 650 [31] guidelines is applied to calculate the section modulus of the stiffening-helix as intermediate stiffener. The wind-structure interaction (WSI) has been studied with the help of Ansys multiphysics-based system coupling of CFX solver for the computational fluid dynamics (CFD) modeling of wind flow and mechanical solver for eigenvalue buckling analysis of tank with the imported wind pressure data from CFD [32,33].

Experimentations and high-end analytical studies on the stiffener performance against wind-induced tank buckling need costly testing, sensor and high-performance computing setup along with significant amount of time. There is a severe scarcity of studies exploring the application of machine learning models for assessing wind pressure variation and buckling of cylindrical thin-walled shell. Gradient boosting regression tree model is found to be efficient for forecasting average aerodynamic pressure and its fluctuation on isolated shells based on their geometric variation, whereas artificial neural network (ANN) model is proved to be effective for the prediction of similar aerodynamic parameters when an elevated cylindrical shell is facing wind-interference scenario due to a nearby building [34,35]. Existing machine learning based studies, which are focused on the shell-buckling, have considered the buckling effect due to axial compressive load, unlike wind pressure induced membrane stress created buckling of thin-walled tank-shell [36–40]. Samaniego et al. [41] and Mishra et al. [42] implemented neural network for solving partial differential equations of linear elastic and nonlinear transient elasto-dynamics problems, which is an efficient amalgamation of computational mechanics and machine learning. These literatures on machine learning applications explores the aerodynamic pressure characteristics and buckling of cylindrical shell in a separate manner. An absence of study focusing on the machine learning model of wind-induced buckling of thin-walled shell is noticed. The authors have also taken up this scope and introduced a novel ANN model which is trained with the data obtained from the WSI analyses of wind-induced buckling of open-top tank strengthened with stiffening-helix. Several analytical parameters are associated with this study, of which the nomenclatures are enlisted in Tab.1.

2 Novelty statement

Wind-induced buckling of large diametric circular steel storage is a fairly under studied area of research. This study provides a fresh approach of addressing this real-life issue of wind-induced buckling damage or failure of such steel-storages, especially tanks, both in terms of innovative stiffening mechanism and procurement of machine learning model for time-economic and easier design of these structures. Importance has been provided to nondimensional parametric study so that generalized understanding can be achieved on the performance of a unique RNA-inspired stiffening-helix to stabilize open-top, ground-supported, cylindrical, steel-tank against wind-induced buckling. This study also addresses the lack of machine learning model for assessing wind-induced shell buckling. An ANN-based prediction model has been introduced for predicting the stability of open-top, cylindrical steel-tank against wind-induced buckling. The proposed stiffening-helix mechanism provides satisfactory stabilization to the tank against wind-induced buckling. Also, the reported accuracy of the predictive ANN model signifies that this model can be used by structural engineers for the safe design of open-top, cylindrical steel-tank against wind-induced buckling.

3 Description of the computational fluid dynamics modeling

3.1 Specifications of virtual model and domain

Bluff body analysis through CFD modeling has been performed to assess the wind pressure. Open top, ground-supported tank model with 20 m diameter (D) has been considered as the bluff body, whose height (H) was varied in a range of 5 to 20 m with 5 m interval, representing for four different H/D ratios (0.25, 0.5, 0.75, and 1.0). The virtual wind domain has been shaped to maintain 3% to 5% blockage ratio, according to the recommendations of architectural institute of Japan (AIJ) [43]. The opening for letting the wind flow in the domain, the domain-roof and walls on the domain-sides are five times the maximum tank dimension (H or D whichever is applicable) apart from the corresponding face of model. The distance between respective model surface and wind outflow opening of the domain is 15 times the extreme dimension of the tank, which provides sufficient space to the vortices generating behind the model, thus avoiding any chance of flow recirculation. Free slip side wall and roof surfaces of the domain do not affect the wind flow. Smooth base of domain and faces of tank represents the surface roughness necessary for open terrain and minimizes the creation of small near-wall vortices around the model, respectively. The schematic diagram of domain showing the dimensions is provided in Fig.1.

3.2 Approach of turbulence modeling

The turbulence modeling of the wind flow has been carried out through Ansys CFX [32]. Despite the reputation of large eddy simulation (LES) model in replicating precise near wall vortices around complex building structures, three-dimensional (3D) unsteady Reynolds averaged Navier Stokes (URANS) model has been applied in this study, as it is fairly accurate and time saving way of simulating sub-grid to large scale vortices around structures with simple geometry like cylindrical tank [44–51]. The formulations of continuity and momentum for 3D URANS turbulence model are provided in Eqs. (1) and (2), which are solved by Reynolds decomposition of incompressible wind flow into fluctuating and time-averaged components [52]. Due to the less accuracy of standard k–ε model, realizable k–ε model has been applied as eddy-viscosity model to define the separation of rotational wind flow’s boundary layer with the expressions of kinetic (k) and dissipated (ε) energies as Eqs. (3) and (4), caused by energy transfer during turbulence in flow [53, 54]. The mean momentum-shift for URANS model, caused by the unsteady convection of flow, is expressed with mean pressure (p) induced isotropic stress, dynamic viscous stress, body force induced mean stress (f_i) and Reynolds stress due to fluctuating velocity components u_i^{\prime } and u_j^{\prime } along ith and jth spatial coordinates. Equational coefficient C₁ of dissipated energy is calculated as Eq. (5). Mean velocity gradient induced kinetic energy (G_k), presented in Eq. (6), is composed of modulus of strain tensor (S) and turbulence viscosity (μ_t), which are showed in Eqs. (7) and (8), respectively. Various equation components C_μ, A_s, {\tilde{\mathrm{\Omega }}}_{ij}, \varphi, and \tilde{S}, are subsequently provided as Eqs. (9)–(13). Average rate of strain tensor (S_ij) is showcased as Eq. (14). Density of air, mean rate of rotation tensor, angular speed, kinematic and dynamic viscosities are presented as ρ, {\overline{\mathrm{\Omega }}}_{ij}, ω_k, ν, and μ. u_i and u_j are mean components of velocity along x_i and x_j. The value of the constants σ_k, C₂, A₀, and σ_ε are 1.0, 1.9, 4.04, and 1.2, accordingly.

(1)\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_i}=0,

(2)\frac{\mathrm{\partial }}{\mathrm{\partial }t}\left(\rho u_i\right)+\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}\left(\rho u_i{u}_j\right)=-\frac{\mathrm{\partial }p}{\mathrm{\partial }{x}_i}+\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}\left(\mu +{\mu }_{\mathrm{t}}\right)\left(\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_j}+\frac{\mathrm{\partial }{u}_j}{\mathrm{\partial }{x}_i}\right),

(3)\frac{\mathrm{\partial }}{\mathrm{\partial }t}\left(\rho k\right)+\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}\left(\rho k{u}_j\right)=\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}\left[\left(\mu +\frac{{\mu }_{\mathrm{t}}}{{\sigma }_{\mathrm{k}}}\right)\left(\frac{\mathrm{\partial }k}{\mathrm{\partial }{x}_j}\right)\right]+G_{\mathrm{k}}-\rho \epsilon ,

(4)\begin{array}{rl}\frac{\mathrm{\partial }}{\mathrm{\partial }t}\left(\rho \epsilon \right)+\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i}\left(\rho \epsilon u_i\right)=& \frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}\left[\left(\mu +\frac{{\mu }_{\mathrm{t}}}{{\sigma }_{\epsilon }}\right)\left(\frac{\mathrm{\partial }\epsilon }{\mathrm{\partial }{x}_j}\right)\right]+\rho C_{1}S\epsilon \\ & -\rho C_2\left(\frac{{\epsilon }^2}{k+\sqrt{\nu \epsilon }}\right),\end{array}

(5)C_1=\mathrm{m}\mathrm{a}\mathrm{x}\left[0.43,\frac{Sk/\epsilon }{Sk/\epsilon +5}\right],

(6)G_{\mathrm{k}}={\mu }_{\mathrm{t}}{S}^2,

(7)S\equiv \sqrt{2S_{ij}{S}_{ij}},

(8){\mu }_{\mathrm{t}}=\rho C_{\mu }\frac{k^2}{\epsilon },

(9)C_{\mu }=\frac{1}{A_0+A_{\mathrm{s}}{\displaystyle \frac{k\sqrt{S_{ij}{S}_{ij}+{\tilde{\mathrm{\Omega }}}_{ij}{\tilde{\mathrm{\Omega }}}_{ij}}}{\epsilon }}},

(10)A_{\mathrm{s}}=\sqrt{6}\mathrm{c}\mathrm{o}\mathrm{s}\varphi ,

(11){\tilde{\mathrm{\Omega }}}_{ij}={\overline{\mathrm{\Omega }}}_{ij}-3{\epsilon }_{ij\mathrm{k}}{\omega }_{\mathrm{k}},

(12)\varphi =\frac{1}{3}{\mathrm{c}\mathrm{o}\mathrm{s}}^{-1}\left(\sqrt{6}\frac{S_{ij},S_{j\mathrm{k}},S_{\mathrm{k}i}}{{\tilde{S}}^3}\right),

(13)\tilde{S}=\sqrt{S_{ij}{S}_{ij}},

(14)S_{ij}=\frac{1}{2}\left(\frac{\mathrm{\partial }{u}_j}{\mathrm{\partial }{x}_i}+\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_j}\right).

3.3 Details of time history analysis and calculation of pressure coefficient

Time history analysis has been performed to assess the aerodynamic pressure of 3.0 s gust duration as per IS 875-2015, part 3 [55], through second order backward Euler method with the converging target value of root mean square (RMS) residual as 0.00001 [56]. The iterative transient analysis has been carried out with 0.001 s time stepping and maximum 10 numbers of iterative cycles at each step, as per AIJ guidelines [43]. Following the works of Portela and Godoy [57] the external (C_pe) and internal (C_pi) coefficients of wind pressure have been assessed with Eqs. (15) and (16). P_{j_{\mathrm{e}}} and P_{j_{\mathrm{i}}} represents the wind pressure measured from circumferential pressure taps on external (j_e) and internal (j_i) surfaces of tank-wall. The pressure taps are located circumferentially on the inside and outside faces of tank-wall at elevations 0.25H, 0.5H, and 0.75H. The tapping point pressure P_j is generally expressed as Eq. (17) as circumferential angle (θ) and time (T) dependent function of measured pressure (P_{{\mathrm{M}}_{\mathrm{n}}}) at nth tapping point. As per the study of Bairagi and Dalui [58], the free stream pressure (P_H) of wind at similar elevation to the tank top is presented as Eq. (18), considering the wind speed as U_H at an elevation equivalent to the tank top. The time-averaged pressure coefficients at the exterior (C_{{\mathrm{p}\mathrm{e}}_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}}}) and interior (C_{{\mathrm{p}\mathrm{i}}_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}}}) surfaces of tank-wall are calculated as Eqs. (19) and (20), from pressure data of N number of time instances.

(15)C_{\mathrm{p}\mathrm{e}}=\frac{P_{j_{\mathrm{e}}}}{P_{\mathrm{H}}},

(16)C_{\mathrm{p}\mathrm{i}}=\frac{P_{j_{\mathrm{i}}}}{P_{\mathrm{H}}},

(17)P_j=P_{M_{\mathrm{n}}}\left(\theta ,T\right),

(18)P_{\mathrm{H}}=0.5\rho U_{\mathrm{H}}^2,

(19)C_{{\mathrm{p}\mathrm{e}}_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}}}=\frac{1}{N}\sum _{j_{\mathrm{e}}=n-1}^{j_{\mathrm{e}}=n}{C}_{\mathrm{p}\mathrm{e}}\left(T\right),

(20)C_{{\mathrm{p}\mathrm{i}}_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}}}=\frac{1}{N}\sum _{j_{\mathrm{i}}=n-1}^{j_{\mathrm{i}}=n}{C}_{\mathrm{p}\mathrm{i}}\left(T\right).

3.4 Validation of computational fluid dynamics model with experimental literature with the help of mesh sensitivity analysis

The present CFD model was already validated with the help of sensitivity analysis of the mesh, in the previous study by the authors, Mukherjee et al. [35], with respect to the wind tunnel test output of Portela and Godoy [59], which is explained in this section. In the studies of Mukherjee et al. [35] full scale tank’s H/D ratio (D = 20 m, and H = 8.75 m) was similar to the scaled model (D = 269.2 mm, and H= 115.7 mm) of Portela and Godoy [59] for wind tunnel test. The basic wind speed in the study of Mukherjee et al. [35] has been taken as 64.8 m/s, which was similar to the scaled speed 19.8 m/s at 116 mm inside wind tunnel test section of the study of Portela and Godoy [59]. The terrain was open terrain for both studies [35,59]. Tetrahedral mesh with inflation at corners and edges for ensuring precise flow separation, has been used for discretization as per the studies of Franke et al. [60]. To determine the optimum sizing of mesh, Mukherjee et al. [35] applied three different meshing sizes, namely fine mesh (FM), medium mesh (MM) and coarse mesh (CM), of which the images and specifications are provided in Tab.2 and Fig.2, respectively. Mukherjee et al. [35] compared the external pressure coefficients in Fig.3(a), obtained from the CFD analyses using the three different mesh sizes, with the same from the experimentation of Portela and Godoy [59]. Similar pressure reporting elevation (0.6H) like the study of Portela and Godoy [59], has been considered by Mukherjee et al. [35]. The percentages of error in pressure coefficient are 3.21%, 14.23%, and 36.63%, accordingly, for FM, MM, and CM mesh sizes in comparison to Portela and Godoy [59], which is shown in Fig.3(b). The FM with the least percentage of error has been determined for the optimum requirement for the mesh sizing. The model and domain have been discretized with the FM, which is exhibited in Fig.4.

3.5 Theoretical validation of computational fluid dynamics model with velocity and turbulence intensity profiles

The theoretical validation was also performed by the authors, Mukherjee et al. [35] in the previous study by comparing wind velocity and turbulence intensity profiles obtained from the CFD model with the theoretical model using power law, as per AIJ [43]. The power laws of wind velocity and turbulence intensity are expressed as Eqs. (21) and (22), in which U_z and I_z are wind speed and turbulence intensity at elevation z, U_{{\mathrm{z}}_{\mathrm{r}\mathrm{e}\mathrm{f}}} is the reference wind speed at reference elevation z_ref, z_G is gradient height of atmospheric boundary and α is coefficient of power law. The study has been performed considering open terrain condition in Indian subcontinent, consequently α and z_G have been taken as 0.188 and 300 m as per IS 875-2015, part 3 [55] and IS SP 64-2001 [61]. The reference wind speed has been considered as 50 m/s at 10 m. The reference turbulence intensity at gradient height has been considered as 1% for CFD model. The comparison of CFD and theoretical profiles for wind velocity and turbulence intensity, taken at 4D distance from the tank as per AIJ [43], is showcased in Fig.5(a) and Fig.5(b), accordingly. Theoretical and CFD profiles of wind speed are nearly similar, whereas some variation is observed for turbulence intensity. Although the variation for turbulence intensity is < 10% upto the elevation similar to the tank’s height, which will not affect the wind pressure distribution pattern in a significant manner. For the present study purpose eight different basic speed of wind has been considered, which are 5, 15, 25, 35, 45, 55, 65, and 75 m/s.

(21)U_{\mathrm{z}}=U_{{\mathrm{z}}_{\mathrm{r}\mathrm{e}\mathrm{f}}}{\left(\frac{\mathrm{z}}{{\mathrm{z}}_{\mathrm{r}\mathrm{e}\mathrm{f}}}\right)}^{\alpha },

(22)I_{\mathrm{z}}=0.1{\left(\frac{\mathrm{z}}{{\mathrm{z}}_{\mathrm{G}}}\right)}^{-\alpha -0.005}.

4 Analogy of wind pressure induced-buckling

4.1 Structural details

Bluff body analyses have been performed through CFD model for eight different basic wind speeds and four various H/D ratios of open-top, cylindrical, steel tank. Six various r/t ratios, which are 750, 1000, 1250, 1500, 1750, and 2000, have been used during the structural modeling of the tanks. The tank-base and top have been considered as fixed and free, respectively. A unique RNA inspired stiffening-helix has been applied as intermediate stiffener along with ring-stiffeners at the top of tank and bottom end of the stiffening-helix. The pitch length (L_P) of the stiffening-helix has been varied four times, which are H/5, 4H/15, 2H/5, and 4H/5. The diagram of the tank with H/D ratio 0.5 is provided in Fig.6, showcasing all the details of stiffening-helix pitch length. The section modulus (Z_TR) of the top wind girder has been designed as the top ring-stiffener design guidelines following API 650 [31], which is shown in Eq. (23). Whereas the stiffening-helix and bottom ring-stiffener section moduli (Z_IR) have been designed as per the intermediate ring-stiffener design method of API 650 [31], which is presented as Eq. (24). The stiffener’s lowest yield strength (F_y) requires to be the smaller value between 210 MPa and strength at peak working temperature (in MPa). The design pressure of wind is P_wd (kPa), expressed as Eq. (25). The design velocity pressure of wind (P_wv) at 10 m elevation and basic pressure (p) have been presented as Eqs. (26) and (27), subsequently. The coefficient of exposure of the pressure of velocity (K_z) is 1.04, the factor of gust (G) is 0.85, the design speed of gust (V) is 190 km/h at 10 m height, the circular tank’s directionality factor of wind (K_d) is 0.95, the factor of structural importance (I) is 1.0 and the topographical factor (K_zt) is 1.0.

(23)Z_{\mathrm{T}\mathrm{R}}=\frac{6H{D}^2}{0.5F_{\mathrm{y}}}\left(\frac{P_{\mathrm{w}\mathrm{d}}}{1.72}\right),

(24)Z_{\mathrm{I}\mathrm{R}}=\frac{6H_{\mathrm{T}\mathrm{I}}{D}^2}{0.5F_{\mathrm{y}}}\left(\frac{P_{\mathrm{w}\mathrm{d}}}{1.72}\right),

(25)P_{\mathrm{w}\mathrm{d}}=P_{\mathrm{w}\mathrm{v}}+0.24,

(26)P_{\mathrm{w}\mathrm{v}}=p{\left(\frac{V_{\mathrm{b}}}{190}\right)}^2,

(27)p=0.00256K_{\mathrm{z}}{K}_{\mathrm{z}\mathrm{t}}{K}_{\mathrm{d}}{V}^{2}IG.

4.2 Element details

Cylindrical steel tank is a thin-walled shell structure, for this reason the tank wall has been discretized with two dimensional (2D) SHELL181 element, which has 4 nodes, each having six degrees of freedom (DOF) [62]. The stiffening-helix and other stiffeners have been meshed with BEAM188 element with two nodes having six DOF each [62]. The stiffener needs to be attached with the steel tank with welded connection as per API 650 [31] and IS 803-1976 [63]. The welded connection between stiffener and tank has been replicated by applying CONTA175 element, as bonded contact with zero penetration using Lagrangian formulation [64–67]. The contact region mesh between tank and stiffeners is shown in Fig.7. The wind pressure from CFD solver has been imported to the mechanical solver and transformed into an asymmetrical surface pressure on the wall of tank by applying SURF154 element with four to eight nodes with three DOF at each node [68]. Similar to the previous study of the authors, Mukherjee et al. [69], 50 mm element size has been used for the wind-induced buckling analysis according to the study of Sun et al. [70], which proposes that the circular tank’s circumference needs to be minimum 1256 times greater than the element size.

4.3 Analytical details

The authors have analyzed the wind-pressure induced buckling of the tank by transforming the wind pressure into an equivalent non-uniform surface pressure, similar to the previous study of the authors, Mukherjee et al. [69,71]. This transformation technique of the wind pressure as arbitrarily distributed surface pressure is inspired from the analytical model of Yang et al. [72,73]. Considering cylindrical coordinate system for cylindrical tank, the surface wind pressure has been decomposed into axial and circumferential pressure components P(z) and P(Φ), respectively, where z and Φ are axial and circumferential axes. The representation of circumferential axis in terms of radius (r) of tank-shell and circumferential angle (θ) is expressed as Eq. (28). The wind pressure as functions of elevation (z) and circumferential angle (θ) are shown as Eqs. (29) and (30). Non-dimensional equation parameters are β₁, β₂, and ϵ (0\le ϵ\le 1). The reference pressure (P_{{\mathrm{z}}_{\mathrm{r}\mathrm{e}\mathrm{f}}}) has been considered as the velocity pressure of wind at the stagnation zone following the study of Yasunaga and Uematsu [74]. In case of analyzing wind-induced nonlinear oscillation of stiffened shell, the eigenvalue problem of the stiffener’s stiffness can be computed by eliminating nonlinear terms of Timoshenko-Ehrenfest beam theory [75]. But, in present analytical approach for wind-induced shell buckling, the buckling stress (P_B) has been formulated through Eq. (31), as the sum of the uniform pressure induced stress (P_U) and the total of all the incremental stresses (P_ŋ) upto ŋth load function. The circumferential compression of the tank-wall creating membrane stress is similar to the behavior of Mindlin-Reissner plate under time-modulated axial force, which in turn produces the circumferential waves of wind-induced buckling [76]. Summation of uniform pressure induced radial deformation ({\delta }_{{\mathrm{r}}_{\mathrm{U}}}) and incremental radial deformations ({\delta }_{{\mathrm{r}}_{\eta }}) caused by the total number (\eta) of load functions express the buckling induced radial deformation ({\delta }_{{\mathrm{r}}_{\mathrm{B}}}) through Eq. (32). The nondimensional longitudinal axis (\zeta) of the cylindrical coordinate system is represented as z/H ratio. The load multiplier (λ) of buckling is expressed as Eq. (33). The governing formulation of eigenvalue buckling analysis is provided in Eq. (34) [33]. Where, stress stiffening matrix, tangent stiffness matrix, buckling load multiplier and eigen vector at mth mode are S_S, K_T, λ_m, and ψ_m.

(28)\Phi =r\theta ,

(29)P\left(\mathrm{z}\right)=P_{Z_{\mathrm{r}\mathrm{e}\mathrm{f}}}\left[1+ϵ\left(\frac{\mathrm{z}}{H}+\frac{1}{2}\right)\right],

(30)P\left(\theta \right)=P_{Z_{\mathrm{r}\mathrm{e}\mathrm{f}}}\left[1+{\beta }_1+ϵ{\beta }_1\left(\mathrm{c}\mathrm{o}\mathrm{s}\theta -1\right)+ϵ{\beta }_2\left(\mathrm{c}\mathrm{o}\mathrm{s}2\theta \right)\right],

(31)P_{\mathrm{B}}=P_{\mathrm{U}}+\sum _{\eta =1}^{\mathrm{\infty }}{ϵ}^{\eta }{P}_{\eta },

(32){\delta }_{r_{\mathrm{B}}}\left(\zeta ,\theta \right)={\delta }_{r_{\mathrm{U}}}+\sum _{\eta =1}^{\mathrm{\infty }}{ϵ}^{\eta }{\delta }_{r_{\eta }}\left(\zeta ,\theta \right),

(33)\lambda =\frac{P_{\mathrm{B}}}{P\left(\mathrm{z},\varphi \right)},

(34)\left[K_{\mathrm{T}}+{\lambda }_m\left[S_{\mathrm{S}}\right]\right]\left\{{\psi }_m\right\}=\left\{0\right\}.

5 Supervised training of artificial neural network

5.1 Details of network architecture

Application of ANN for the predictive modeling of shell buckling is limited to the effect of axial compression [36]. The authors have introduced a novel feed-forward backpropagation ANN model for prediction of tank stability against wind pressure-induced buckling through training with the WSI data. Similar to the physical nervous system with neurons, the ANN consists of computational units called neuron. Similar to the information receiving, transfer, process and command delivering mechanism of physical nervous system, ANN receives the data as input, transfers through hidden layers, processes with neurons and delivers the computational result as output. The input variables of present study are H/D, r/t, L_P/H, and V_b (m/s) to assess the stability of tank against wind-induced buckling in terms of load multiplier (λ) of buckling as output. Neural network can be improvised with particle swarm optimization or devised with higher number of hidden layers and neurons to assess highly nonlinear and complex shell buckling with multi-input and multi-output [77]. But when the data complexity is relatively less with single output, single hidden layer can be sufficient, which is also applied in present study [78]. To finalize the optimum neuron requirement, variations have been carried out in the neuron number from ten to 40. The architecture of ANN model is provided in Fig.8, where ṋ is the neuron number.

5.2 Performance assessing parameters of the network

Similar to the previous study of Mukherjee et al. [35], the authors used regression or correlation coefficient (R) and mean squared error (MSE) as the performance assessing parameters for the ANN model, which are shown in Eqs. (35) and (36). The regression value ranges from −1.0 to 1.0. R value closer to 1.0 shows the higher precision level of correlation performance. In case of MSE value, the smaller the better in terms of accuracy and minimization of error. The number of training data set, the target output and ANN output of the lth data set are η, t_l and a_l. For the activation purpose the sigmoid function has been used, which is provided in Eq. (37). The output of ANN has been expressed as a function of input p_i, weight W and bias b, which is shown in Eq. (38). The weight matrix (W) is expressed as Eq. (39), where number of neuron and input are represented as ṋ and p_l, respectively. Due to the adaptability, quick convergence and optimum efficiency, Levenberg–Marquardt (LM) training function has been applied for the data training [57,79]. The training parameters of LM algorithm are adjustment parameter (mu), maximum value (mu_max), increment factor (mu_inc) and decrement factor (mu_dec) of adjustment parameter by Marquardt. When the gradient is less than minimum gradient (min_grad) or failure number of validation is more than maximum failure number in validation (max_fail), the training gets stopped.

(35)R=\sqrt{1-\frac{{\displaystyle \sum _{l=1}^{\eta }{\left(t_l-a_l\right)}^2}}{{\displaystyle \sum _{l=1}^{\eta }{\left(t_l-\overline{t_l}\right)}^2}}},

(36)MSE=\frac{1}{\eta }\sum _{l=1}^{\eta }{\left(t_l-a_l\right)}^2,

(37)y=\frac{2}{1+{\mathrm{e}}^{-2x}}-1,

(38)a=f\left(W{p}_{\mathrm{i}}+b\right),

(39)W=\left[\begin{array}{cccc}{w}_{1,1}& w_{1,2}& \dots & w_{1,p_l}\\ w_{2,1}& w_{2,2}& \dots & w_{2,p_l}\\ \dots & \dots .& \dots & \dots \\ w_{ṋ,1}& w_{ṋ,2}& \dots & w_{ṋ,p_l}\end{array}\right].

6 Results and discussions

6.1 Wind-induced buckling

The circumferential distribution of mean pressure coefficients of wind on external and internal surfaces of tank-wall, which are calculated from the CFD analyses, are expressed graphically from Fig.9(a)–Fig.9(d), for tank’s H/D ratio 0.25 to 1.0, accordingly. Positive wind pressure and suction are observed at the frontal region and side to leeward zone, respectively. The positive pressure and suction at the external wall-surface rises with the incremental pattern of the ratios of H/D and tapping elevation (z) to tank height (H). The internal suction subsequently decreases and increases with the rise in z/H and H/D ratios. In the earlier studies, the authors described the wind-induced buckling mechanism of open-top steel-tank [69,71]. The membrane stresses occur in the tank-wall and propagate through the sides of the tank before meeting at the frontal zone due to the peripheral distribution of aerodynamic pressure, which creates circumferential buckling waves at the frontal region’s wall. The distribution of wind pressure and consequent buckling waves are exhibited schematically in Fig.10(a) and 10(b), respectively. The radial deformations of the tanks for the considered cases of wind-induced buckling have been analyzed. The radial deformation helps to assess the damaged portion of the tank-wall due to the buckling waves caused by wind-induced membrane stress. The radial deformation has been normalized with respect to the maximum value. The normalized radial deformation contours for tanks with four different H/D ratios are presented in Fig.11(a)–Fig.11(d). The outward and inward radial deformations have been regarded as positive and negative, accordingly in the contour, which is shown in the color legend. The {\delta }_{{\mathrm{r}}_{\mathrm{B}\mathrm{N}}} contour exhibits presence of buckling waves at the wind-ward region of the tank wall. The number and amplitude of the buckling waves respectively rises and minimizes with the decrement in the L_P/H ratio of the stiffening-helix. Greater slenderness ratio of the tank shows larger amplitude and number of buckling waves. It is observed that for a particular H/D ratio of the tank with stiffening-helix having a certain L_P/H ratio, the resulting normalized radial deformation ({\delta }_{{\mathrm{r}}_{\mathrm{B}\mathrm{N}}}) of tank-wall and number of buckling waves due to wind-induced buckling, is nearly identical in all the combinations of wind speed and wall-thickness cases. This observation signifies that although normalized radial deformation is useful for damage assessment, it is not a reliable parameter of tank’s stability against wind-induced buckling.

The buckling load multiplier (λ) for all the cases of open-top steel tanks with stiffening-helix, have also been analyzed, which are presented in Fig.12. Different values of load multiplier for each case of wind-induced tank-buckling signifies that the buckling load multiplier is a reliable parameter to assess the stability of a tank. Consequently, analysis of load multiplier is a way of understanding the performance of stiffening-helix against wind-induced tank buckling. The multiplier of buckling load reduces by 2.54%, 2.23%, 2.63%, 2.04%, 1.78%, 1.53%, and 6.6% with the gradual rise in basic wind speed from 5 to 75 m/s. With the decrease in L_P/H ratio from 4/5 to 1/5, the λ value increases by 79.35%, 59.5%, and 63.4%, accordingly. Gradual increment in r/t ratio from 750 to 2000 reduces the load multiplier value by 50.6%, 42.25%, 36.16%, 31.62%, and 28.41%, accordingly. The λ value decreases by 68.07%, 32.93%, and 17.96%, subsequently for the gradual increase in slenderness ratio from 0.25 to 1.0. As per pressure vessel design guidelines of Moss [80], the safe value of buckling load multiplier should be between 2 to 3. Based on this guideline five distinct stability conditions of tank have been identified in the present study, which are, instability (λ < 1.0), critical stability (λ = or \approx 1.0), low stability (), safe/sufficient stability () and high stability (\lambda \ge 3.0). Instability condition is observed for tank with H/D ratio 0.5, 0.75, and 1.0, for r/t ratio \ge 2000, \ge 1750, \ge 1500, when L_P/H ratio is \ge 1/5, respectively.

6.2 Performance of artificial neural network model

The ANN model has been trained with 20%, 40%, 60%, and 80% of 768 data sets obtained from the multiphysics analyses of wind-induced tank-buckling, of which the R and MSE are presented graphically in Fig.13(a)–Fig.13(d), accordingly. Here Y is the predicted output and T is the actual output. The key aspects of training process, such as, tuning of hyperparameter, time duration of training, etc. have been addressed carefully to minimize the risk of model degradation. To lessen the probability of overfitting, single hidden layer of the neural network has been used. Variation in percentage of data and number of neurons have been applied while training to determine the optimum network, which can address the computational complexity, and challenges in gradient propagation. The best performance based on MSE and R is observed for 20 neurons with 80% data set at 63rd epoch. Whereas, 30 neurons with 60% data set provides sufficient performance at 54th epoch.

After determining the optimum architecture of the neural network, random tests have been carried out to predict load multipliers of wind-induced buckling of an open-top tank with H/D ratio 1.25 under basic wind velocity 45 m/s, considering all the combinations of r/t ratio and L_P. The random test results of buckling load multiplier, produced by proposed ANN model, have been compared with the same obtained through the WSI analyses on similar configuration of tank under V_b 45 m/s, to escalate the proposed ANN model’s accuracy, which featured in Fig.14. 0.61%, 1.57%, 0.3%, and 1.4% error of ANN model has been noticed with respect to WSI analyses for L_P/H ratio 4/5, 2/5, 4/15, and 1/5, respectively. The contours of {\delta }_{{\mathrm{r}}_{\mathrm{B}\mathrm{N}}}of the tank with H/D ratio 1.25 for each test case of L_P/H ratio, are accordingly provided in Fig.15(a)–15(d), which have been obtained from the WSI analyses. Fig.14 and Fig.15 signifies that in case of H/D ratio 1.25, open-top, cylindrical, steel tank is safe (2 < λ < 3) against wind-induced buckling under 45 m/s basic speed of wind, when r/t and L_P/H ratios are 1000 and 4/15, respectively.

7 Conclusions

It is observed that windward region is the most affected portion of a tank against wind-induced buckling. Nondimensional radial deformation is very much useful parameter to escalate the damage of the tank-wall due to wind-induced buckling. Load multiplier is the most reliable parameter for assessing the stability of tank against wind-induced buckling.

This study highlights that RNA-inspired stiffening-helix can be an effective alternative to ring-stiffener for providing capacity against wind-induced buckling to cylindrical steel tanks. It is reported that wind-induced buckling capacity of open-top, cylindrical, ground-supported, steel tank increases with the decrease in basic wind speed (V_b), H/D, r/t, and L_P/H ratio.

This study can be helpful to determine which nondimensional specifications (H/D, r/t, and L_P/H) are necessary for the safe and economic construction of a tank strengthened with stiffening-helix based on the basic wind speed (V_b) region, in which that tank need to be constructed.

Considering most severe wind speed situation, the most economic while safe combination of r/t and L_P/H ratio is 2000 and 4/15, respectively for tank’s H/D ratio 0.25. Similarly, under 75 m/s wind speed, safety with economy can be provided to tanks with H/D ratio 0.5, 0.75, and 1.0, for r/t ratio 1750, 1500, and 1250, when the L_P/H ratio is 1/5, respectively.

With 99.11% accuracy, the proposed ANN model can greatly aid the structural engineers to design open-top, cylindrical, ground-supported, steel tank considering wind-induced buckling by providing load multiplier values in very less amount of time compared to the cumbersome and too much time-consuming WSI analyses.

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