# An Improved Winkler Foundation Modulus for a Beam in a Full Space
###### Abstract
[PERSON] [16] proposed the formula of the Winkler foundation modulus by improving the fitting method of [PERSON] [14] based on the assumption that the overall distribution of displacements, bending moments, and shear forces was consistent from the two methods. [PERSON] and [PERSON] [17] proposed the equivalent formula of the Winkler foundation modulus for a slab. [PERSON] et al. [18] compared the closed-form solutions of pipeline responses under tunneling based on the elastic continuum model and elastic foundation beam model. The results indicated that the modulus of subgrade is two times or even more than that proposed by [PERSON]. Given that the underground structures are typically influenced by the free-field displacement, [PERSON] et al. [19; 20] obtained the Winkler foundation modulus for a pipe embedded in the ground with finite burial depth and infinite burial depth under free-field soil displacement using [PERSON]'s fitting method. Most of the theoretical studies mentioned above were mainly concerned with the Winkler foundation modulus for a beam on the surface. In many cases, underground structures are embedded at great depths. The Winkler modulus of a semi-infinite space is not applicable to solving this problem. Although [PERSON] et al. [20] proposed the formula of the full-space Winkler foundation modulus using a theoretical method, there are limitations in the research. When the beams and foundation are inseparable, there are two kinds of interaction between the beam and ground, including normal interactions and tangential interactions. The traditional elastic foundation beam model ignores the tangential interaction between a structure and the ground and only considers the normal interaction. [PERSON] et al. [21; 22] improved the traditional elastic foundation beam model by considering the tangential interaction, wherein the normal and tangential interactions are solved independently, as represented by the Winkler foundation spring and rotation spring. In other words, when solving for tangential interactions, the effects of normal interactions should be ignored. Similarly, the effects of tangential interactions should be ignored when solving for normal interactions. In [PERSON]'s method [20], the bending responsesof an infinitely long beam on the Winkler foundation were derived without considering the tangential interaction between the beam and ground. The beam bending responses of an Infinitely long beam on the three-dimensional elastic foundation were obtained by taking into consideration the tangential interaction between the beam and the ground. The formula of the Winkler foundation modulus was then obtained by establishing a connection between the solutions from these two methods through fitting data. During the process of obtaining the Winkler foundation modulus, the tangential interaction is considered in [PERSON]'s method. Thus, an improved Winkler foundation modulus should be proposed by ignoring the tangential interaction between the beam and soil in the elastic foundation beam model and the three-dimensional elastic continuum model. In this paper, an improved expression of the Winkler modulus for a beam embedded in a full-space foundation is derived. First, the bending responses of the beam on the Winkler foundation and in the three-dimensional elastic foundation are obtained, respectively, wherein, the standard displacement is applied on the beams in the two methods and the tangential interaction at the beam-ground interface is ignored. Then, the formula for the Winkler foundation modulus is obtained by fitting the bending response from the two methods. Finally, the effectiveness of the Winkler foundation modulus is verified by comparing the results with numerical results and previous studies.