Atomic force microscopy utilizes one of these structures with a sharp tip at one end.
The classical differential equation used to analyze these structures can be given as "E I" times "d-to-the-fourth w" over "d  x-to-the-fourth" equals q. An improved model of these structures that takes into account shear deformation was developed  by Timoshenko. In the classical theory of these structures, their flexural rigidity is defined as their Young's modulus times  their moment of inertia. That theory of these structures is based on the assumption that the plane normal to the neutral  axis before deformation remains normal to the neutral axis after deformation and is named for (*) Euler and Bernoulli.  Atomic force microscopy utilizes one of these structures with a sharp tip at one end. Instead of resisting twisting or axial effects,  these elements are designed to resist bending effects. For 10 points, name these basic structural components whose lengths are  much larger than their thicknesses and widths, and which come in an "I" form.
The classical differential equation used to analyze these structures can be given as "E I" times "d-to-the-fourth w" over "d  x-to-the-fourth" equals
cantilever beam [or fixed-free beam; or fixed-fixed beam; or guided beam; accept other reasonable equivalents that include beam, such as I-beam; prompt on partial answers that do not include beam]