An exact deflection solution to a type of cantilever with partially built-in end using strong boundary conditions

Abstract

The problem of cantilever deflection is a classic worked example in most elasticity textbooks. Due to the inability for the displacement field to completely satisfy the boundary condition of zero displacement at the entire built-in end from the top surface to the bottom surface, the best solution so far is attained from weak boundary conditions. This paper proposes a modified problem in which the boundary is partially built-in, i.e. the cantilever is clamped on the top and bottom but free on the other two sides. Therefore, the cantilever’s displacement is prohibited only at the top and bottom surfaces at the partially built-in end. By implementing zero displacement therein, the displacement solution can be obtained based on strong boundary conditions. Furthermore, the condition of zero horizontal displacement at the mid-point of the partially built-in end is also achieved through the prescribed boundary conditions. Results also reveal that the displacement at the partially built-in end can be reduced by selecting materials with negative Poisson’s ratio. As an alternative to being a worked example, the proposed problem can be used as an assessment question.