Keywords

1 Introduction

The nonlinear response of a practical structure is affected by various factors such as system properties, operational coefficients, loads and environment, as well as uncertainties in collected information and estimation models. Liu et al. investigated the random mean values of elastoplastic responses of structure using a probabilistic partial differentiation approach [1]. Feng et al. introduced a stochastic elastoplastic analysis of two-dimensional engineering structures with the aid of sampling-based machine learning algorithm [2]. In our study, the uncertain parameters were studied simultaneously within the nonlinear dynamical framework with the help of an advanced machine learning (ML) technique [3]. By using the ML algorithm, an explicit regression function can be obtained to represent the relationship between the uncertain inputs and the nonlinear responses. Subsequently, frequent, and fast nonlinear prognosis can be conducted to assess the nonlinear behavior of engineering structures during the dynamic loading process.

Here, a brief introduction to the proposed ML-aided framework for nonlinear dynamics of engineering structure is given. The two main components of the approach are briefly introduced. First, the deterministic solution to geometric–elastoplastic dynamics is presented. Then, the novel ML technique named the “extended support regression” is introduced. To demonstrate the accuracy and applicability of the proposed framework, an illustrative numerical case is incorporated to build the proposed framework and demonstrate the nonlinear response for the concerned structure.

2 Methods

2.1 Solution to Geometric–Elastoplastic Dynamics

For structural systems with both material and geometric nonlinearities, the plastic strain and second-order Green–Lagrange terms must be considered in the incremental strain–displacement relations as:

$$ \Delta {{\varvec{\upvarepsilon}}} = \Delta {{\varvec{\upvarepsilon}}}_{e} + \Delta {{\varvec{\upvarepsilon}}}_{p} + \Delta {{\varvec{\upvarepsilon}}}_{g} = ({\mathbf{B}} + {\mathbf{B}}_{g} )\Delta {\mathbf{u}} $$
(1)

where \(\Delta {{\varvec{\upvarepsilon}}}_{e}\), \(\Delta {{\varvec{\upvarepsilon}}}_{p}\) and \(\Delta {{\varvec{\upvarepsilon}}}_{g}\) denote the elastic, plastic and high-order strain increments; \({\mathbf{B}}\) and \({\mathbf{B}}_{g}\) denote the material and geometric nonlinear strain–displacement matrixes of the deformation. In the dynamics framework without the effects of external load and damping, the equilibrium function of a nonlinear structural domain by using the principle of virtual work can be represented as:

$$ \int_{V} {\delta {{\varvec{\upvarepsilon}}}^{T} {{\varvec{\upsigma}}}} dV - \int_{S} {\delta {\mathbf{u}}^{T} {{\varvec{\uptau}}}} dS + \int_{V} {\delta {\mathbf{u}}^{T} \rho {{\ddot{\mathbf u}}}} dV = 0 $$
(2)

where \({{\ddot{\mathbf u}}}\) denotes the virtual acceleration vector; \(\rho\) denotes the material density and \({{\varvec{\uptau}}}\) denotes the surface traction along the boundary. By following the incremental strategy, and substituting Eq. (1) into the above virtual field, it can be re-expressed as:

$$ \begin{aligned} \left( {\int_{V} {{\mathbf{B}}^{T} {\mathbf{D}}_{{ep}} {\mathbf{B}}} dV + \int_{V} {{\mathbf{B}}_{g}^{T} {\mathbf{DB}}_{g} } dV} \right)\Delta {\mathbf{u}} & = \int_{S} {{\boldsymbol{\Phi }}^{T} \;{\boldsymbol{\tau }}} dS - \int_{V} {({\mathbf{B}}} + {\mathbf{B}}_{g} )^{T} {\boldsymbol{\sigma }}dV \\ & \quad - \int_{V} {{\boldsymbol{\Phi }}^{T} \rho {\boldsymbol{\Phi }}dV{{\ddot{\mathbf u}}}} = 0 \\ \end{aligned} $$
(3)

Consequently, the global governing equation of the dynamic geometric-material hybrid nonlinearities problem can be represented as:

$$ ({\mathbf{K}}_{ep} + {\mathbf{K}}_{g} )\Delta {\mathbf{U}} = {\mathbf{F}}_{ex} - {\mathbf{R}}_{in} - {{M\ddot{\mathbf U}}} $$
(4)

By using the Newmark time integration technique, the solution to the above equation can be acquired through:

$$ \left\{ \begin{gathered} (a_{0} {\mathbf{M}} + {\mathbf{K}}_{ep}^{t} + {\mathbf{K}}_{g}^{t} )\Delta {\mathbf{U}}^{t + 1} = {\mathbf{F}}_{ex}^{t + 1} - {\mathbf{R}}_{in}^{t} + (a_{0} {\mathbf{U}}^{t} + a_{1} {\dot{\mathbf{U}}}^{t} + a_{2} {{\ddot{\mathbf U}}}^{t} ){\mathbf{M}} - a_{0} {\mathbf{MU}}^{t} \hfill \\ a_{0} = {1 \mathord{\left/ {\vphantom {1 {\left[ {\beta (\Delta t)^{2} } \right], \, a_{1} = {1 \mathord{\left/ {\vphantom {1 {(\beta \Delta t), \, }}} \right. \kern-0pt} {(\beta \Delta t), \, }}a_{2} = {1 \mathord{\left/ {\vphantom {1 {(2\beta ) - 1}}} \right. \kern-0pt} {(2\beta ) - 1}}}}} \right. \kern-0pt} {\left[ {\beta (\Delta t)^{2} } \right], \, a_{1} = {1 \mathord{\left/ {\vphantom {1 {(\beta \Delta t), \, }}} \right. \kern-0pt} {(\beta \Delta t), \, }}a_{2} = {1 \mathord{\left/ {\vphantom {1 {(2\beta ) - 1}}} \right. \kern-0pt} {(2\beta ) - 1}}}} \hfill \\ \end{gathered} \right. $$
(5)

where \(\Delta t\) denotes the incremental time step; \(\beta = {1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4}\) and \(\kappa = {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}\) are set for all the calculations based on the trapezoidal rule [4].

2.2 Extended Support Vector Regression

After thousands of simulations of deterministic nonlinear dynamic analysis based on the finite element model, the collected structural response was considered as the training dataset for the ML technique to recognize the relation between outputs and input parameters.

The adopted extended support regression was the ML technique proposed by Feng et al. [5]. It is based on the theory of classical support vector machine (SVM) and the doubly regularized SVM (DrSVM), and it offers notable practicability in handling nonlinear prediction problems. For detailed definitions of each parameter in Eq. (6), see Feng et al. [6].

$$ \hat{f}_{N} ({\mathbf{x}}) = ({\mathbf{p}}_{k} - {\mathbf{q}}_{k} )^{T} {\hat{\mathbf{k}}}({\mathbf{x}}) - {\hat{\mathbf{e}}}_{k}^{T} {\hat{\mathbf{G}}}_{k} {{\varvec{\upupsilon}}}_{k}^{*} $$
(6)

where \({\mathbf{p}}_{k} ,{\mathbf{q}}_{k}\) denote two positive kernelized parameters; \({{\hat{\mathbf k}(x)}}\) denotes the kernel matrix; \({\hat{\mathbf{e}}}_{k} ,{\hat{\mathbf{G}}}_{k}\) denote two matrix vectors; \({{\varvec{\upupsilon}}}_{k}^{*}\) denotes the obtained solution.

2.3 Modelling

2.3.1 Geometric–Elastoplastic Dynamics of Transmission Tower

As a numerical example, the nondeterministic nonlinear behaviors of a 3D transmission tower structure were investigated, and the geometry is presented in Fig. 1. In this case, the four supports were fixed and the magnitude of a time-varying pressure \(P(t) = 8e10 \times (1 - \cos (20t))\) Pa, was applied at the tip of the tower. By establishing the surrogate system, the nonlinear responses of the tower were predicted at various time steps. The uncertainty information of the system is listed in Table 1.

Fig. 1
Three 3-D transmission tower structures, along with their geometry, indicate P, A, a vertical height of 55 meters, and a width of 10 meters.

Transmission tower geometry and adopted mesh condition

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Table 1 Variational input data of transmission tower
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By substituting the new input information into the X-SVR model, the predicted contour plots of nonlinear deflections of the tower structure at three time steps are shown in Fig. 2, as well as the deterministic numerical simulation results as verification. In Fig. 2, the predicted nonlinear displacements from X-SVR correlate well with the numerical simulation results.

Fig. 2
Four contour plots of nonlinear deflections of the tower structure at t equal 0.25 and 0.50 seconds for X-S V R predicted and numerical. They indicate the regions of Y-displacement in v over L using color codes. A color chart ranging the values of Y-displacement in v over L is given near the simulations.

X-SVR predicted and numerical simulated nonlinear deflection of a transmission tower at different times

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3 Conclusions

In the numerical modelling example, the nonlinear response of the transmission tower predicted by the proposed method was compared with the numerical simulation result and satisfactory alignment was identified from the comparison. Consequently, the new nonlinear dynamic analysis framework aided by the ML technique for engineering structures was verified as effective for practical engineering problems. We believe that the proposed framework can improve the accuracy and efficiency of the relevant structural nonlinear dynamic’s evaluation process.