The elastic foundation beam method is usually used to analyze the soil-structure interaction. For elastic foundation beams under uniform loads and simple boundary conditions

their theoretical solutions can be obtained from the previous research. Piles under lateral loads and embedded retaining walls are generally analyzed according to the theory of beam on elastic foundation and their coefficients of subgrade reaction are assumed to increase with the increase of the depth. For elastic foundation beams with linearly distributed coefficient of subgrade reaction or with uniform coefficient of subgrade reaction but complex boundary conditions

it is difficult to obtain their theory solutions. In this case

finite difference method or finite element method is used to obtain their approximate solution. When the beam on elastic foundation with linearly distributed coefficient of subgrade reaction is analyzed by finite element method

the calculation accuracy is not ideal if the number of elements are not enough. A 5 times polynomial displacement function and corresponding stiffness matrix of beam element with 2 nodes are presented by adopting weighted residual method. An approximate solution of the deflection of beams under linear distribution of the force is expressed with quintic polynomial. And then more precise nodal displacements and the distribution of element displacements are achieved by the fewer elements. Therefore

the computational efficiency of the presented method is improved. Furthermore

the postprocessing is simplified because the internal force distributions of element can be derived from the displacement function.