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An accurate approach to determining the spatiotemporal vehicle load on bridges based on measured boundary slopes

Alemdar Hasanov, Onur Baysal

TL;DR

The paper tackles the inverse problem of identifying a spatiotemporal vehicle load $F(x,t)$ on a long bridge from end-slope measurements, within a damped Euler-Bernoulli beam framework. It develops forward analysis for a nonhomogeneous beam with internal damping and Kelvin-Voigt damping, and then formulates a Tikhonov-regularized inverse problem using input-output operators $\Phi_0$ and $\Phi_\ell$ tied to boundary slopes $\theta_0(t)$ and $\theta_\ell(t)$. A key contribution is the explicit adjoint-based gradient $J'(F)=\varphi(x,t;F)$, with a well-posed backward problem and a quantified Lipschitz constant $L_G=\sqrt{(e^{T}-1)/(2\kappa_0)} \; \ell^2 \; C_0 \; C_1$, ensuring monotone convergence of gradient methods. The authors establish compactness of the input-output maps, a priori and trace estimates for the forward problem, and existence of a quasi-solution, enabling robust, low-cost vehicle-load identification from boundary data with practical implications for bridge safety and monitoring.

Abstract

In this paper, a novel mathematical model is developed to evaluate the spatiotemporal vehicle loads on long bridges from slope measurements made at the ends of a bridge based on Euler-Bernoulli beam model with internal and external damping. The mathematical modelling of this phenomena leads to the inverse source problem of determining the spatiotemporal vehicle load $F(x,t)$ in the variable coefficient Euler-Bernoulli equation $ρ_A(x)u_{tt}+μ(x) u_{t}+(r(x)u_{xx})_{xx}+(κ(x)u_{xxt})_{xx}=F(x,t)$, $(x,t)\in Ω_T:=(0,\ell)\times (0,T)$ subject to the "simply supported" boundary conditions $u(0,t)=(r(x)u_{xx}+(κ(x)u_{xxt})_{x=0}=0$, $u(\ell,t)=(r(x)u_{xx}+(κ(x)u_{xxt})_{x=\ell}=0$, from the both measured outputs: $θ_1(t):=u_x(0,t)$ and $θ_2(t):=u_x(\ell,t)$, that is, the measured boundary slopes. It is shown that the input-output maps $(ΦF)(t):=u_x(0,t;F)$, $(ΨF)(t):=u_x(\ell,t;F)$, $F \in \mathcal{F}\subset L^2(Ω_T)$, corresponding to the inverse problem, are compact and Lipschitz continuous. Then Tikhonov functional $J(F)=\Vert ΦF-θ_1 \Vert_{L^2(0,T)}^2+\Vert ΨF-θ_2 \Vert_{L^2(0,T)}^2$ is introduced to prove the existence of a quasi-solution to the inverse problem. An explicit gradient formula for the Fréchet derivative of the Tikhonov functional is derived. The Lipschitz continuity of the Fréchet gradient, which guarantees the monotonicity of iterations in gradient methods, has been proven.

An accurate approach to determining the spatiotemporal vehicle load on bridges based on measured boundary slopes

TL;DR

The paper tackles the inverse problem of identifying a spatiotemporal vehicle load on a long bridge from end-slope measurements, within a damped Euler-Bernoulli beam framework. It develops forward analysis for a nonhomogeneous beam with internal damping and Kelvin-Voigt damping, and then formulates a Tikhonov-regularized inverse problem using input-output operators and tied to boundary slopes and . A key contribution is the explicit adjoint-based gradient , with a well-posed backward problem and a quantified Lipschitz constant , ensuring monotone convergence of gradient methods. The authors establish compactness of the input-output maps, a priori and trace estimates for the forward problem, and existence of a quasi-solution, enabling robust, low-cost vehicle-load identification from boundary data with practical implications for bridge safety and monitoring.

Abstract

In this paper, a novel mathematical model is developed to evaluate the spatiotemporal vehicle loads on long bridges from slope measurements made at the ends of a bridge based on Euler-Bernoulli beam model with internal and external damping. The mathematical modelling of this phenomena leads to the inverse source problem of determining the spatiotemporal vehicle load in the variable coefficient Euler-Bernoulli equation , subject to the "simply supported" boundary conditions , , from the both measured outputs: and , that is, the measured boundary slopes. It is shown that the input-output maps , , , corresponding to the inverse problem, are compact and Lipschitz continuous. Then Tikhonov functional is introduced to prove the existence of a quasi-solution to the inverse problem. An explicit gradient formula for the Fréchet derivative of the Tikhonov functional is derived. The Lipschitz continuity of the Fréchet gradient, which guarantees the monotonicity of iterations in gradient methods, has been proven.
Paper Structure (7 sections, 11 theorems, 76 equations, 1 figure)

This paper contains 7 sections, 11 theorems, 76 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The mathematical model of bridge-vehicle load interaction
  3. Analysis of the forward problem and a'pripori estimates
  4. The input-output operators
  5. The Tikhonov functional and existence of a quasi-solution
  6. Fréchet derivative of the Tikhonov functional
  7. The Lipschitz continuity of the Fréchet gradient and monotonicity of iterations

Key Result

Theorem 1

Assume that the basic conditions (3) hold. Then for the weak solution $u\in L^2(0,T;\mathcal{V}^2(0,\ell))$ with $u_t\in L^2(0,T;L^2(0,\ell))$, $u_{tt}\in L^2(0,T;H^{-2}(0,\ell))$ of the forward problem (1), the following estimates are satisfied: where $\mathcal{V}^2(0,\ell):=\{v\in H^2(0,\ell):~ v(0)=v(\ell)=0\}$, and $\rho_0,r_0>0$ are the constants introduced in (3).

Figures (1)

  • Figure 1: Geometry of the spatiotemporal vehicle load on a long bridge based on the damped Euler-Bernoulli beam model

Theorems & Definitions (11)

  • Theorem 1
  • Corollary 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Theorem 2
  • Lemma 4
  • Lemma 5
  • Theorem 3
  • Theorem 4
  • ...and 1 more