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

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.