Normal spring constants of cantilever plates for different load distributions and static deflection with applications to atomic force microscopy

The normal spring constant describes how a cantilever plate deflects under an applied load. The application of microcantilevers in atomic force microscopy (AFM) and in other technologies employing such cantilevers as probes has sparked a significant interest in the knowledge and the determination of their spring constants. Common to most applications involving microcantilevers is the measurement of the deflection at their free end. In AFM a tip attached close to the end probes the forces on the cantilever and therefore the spring constant of a cantilever point-loaded at its free end is of high interest. In case the whole cantilever is used as a sensor, the deflection measured at the end depends on the load distribution over the entire surface area and other spring constant values are relevant. In this article, we derive analytical expressions for the normal spring constants of rectangular, triangular, picket, and V-shaped cantilevers under different load distributions. The expressions derived are for end-loaded, homogeneously loaded, and uniformly varying loaded cantilevers. From the analytical expressions the spring constants can be determined and converted into each other rendering the need to recalibrate unnecessary if the load distribution is changed.