Question #43912

A bimetallic cantilever strip of length $L$ and uniform flexural rigidity $EI$ is heated such that the temperature difference between its top and bottom surfaces is $\Delta T(x) = \Delta T_0 \left(\frac{x}{L}\right)$, where $x$ is the distance from the fixed end ($x=0$). This temperature difference induces a local curvature $\frac{d^2y}{dx^2} = \beta \Delta T(x)$, where $\beta$ is a constant depending on the material properties. The effective spring constant $k_{eff}$ of the strip at its free end is defined by equating the total elastic strain energy stored in the strip to the potential energy of an equivalent spring $\frac{1}{2}k_{eff}y_{max}^2$, where $y_{max}$ is the transverse deflection at the free end. Assuming small deflections, what is the value of $k_{eff}$?