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

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 $\rho_A(x)u_{tt}+\mu(x) u_{t}+(r(x)u_{xx})_{xx}+(\kappa(x)u_{xxt})_{xx}=F(x,t)$, $(x,t)\in \Omega_T:=(0,\ell)\times (0,T)$ subject to the "simply supported" boundary conditions $u(0,t)=(r(x)u_{xx}+(\kappa(x)u_{xxt})_{x=0}=0$, $u(\ell,t)=(r(x)u_{xx}+(\kappa(x)u_{xxt})_{x=\ell}=0$, from the both measured outputs: $\theta_1(t):=u_x(0,t)$ and $\theta_2(t):=u_x(\ell,t)$, that is, the measured boundary slopes. It is shown that the input-output maps $(\Phi F)(t):=u_x(0,t;F)$, $(\Psi F)(t):=u_x(\ell,t;F)$, $F \in \mathcal{F}\subset L^2(\Omega_T)$, corresponding to the inverse problem, are compact and Lipschitz continuous. Then Tikhonov functional $J(F)=\Vert \Phi F-\theta_1 \Vert_{L^2(0,T)}^2+\Vert \Psi F-\theta_2 \Vert_{L^2(0,T)}^2$ is introduced to prove the existence of a quasi-solution to the inverse problem. An explicit gradient formula for the Fr\'{e}chet derivative of the Tikhonov functional is derived. The Lipschitz continuity of the Fr\'{e}chet gradient, which guarantees the monotonicity of iterations in gradient methods, has been proven.