# An Improved Winkler Foundation Modulus for a Beam in a Full Space
###### Abstract  
The Winkler foundation modulus is key to evaluating the response of underground structures using the elastic foundation beam model. In this paper, an improved formula of the Winkler foundation modulus for a beam embedded in a full space is proposed to overcome the limitation of inconsistent assumptions in previous studies. To achieve this goal, the bending responses of the beam are obtained using the elastic foundation beam model and three-dimensional elastic continuum model, respectively, wherein a consistent assumption is proposed that tangential interactions at the beam-ground interface are ignored in the two models. In addition, as deformation of the site is an important source of the underground structure response, the beam is applied to standard soil displacement of the free field on the Winkler foundation to improve the accuracy of the Winkler modulus obtained by fitting solutions based on the concentrated force on the beam. The formula for the Winkler foundation is obtained by equating the first zero of the bending moment in the two models. The Winkler foundation modulus is verified by comparing the results with numerical solutions and previous studies. elastic foundation beam model; underground structures; Winkler foundation modulus; full space +
Footnote †: journal:  
0000-0002-1412-2277-1000/0000-00002-1412-22700/00000-0000-0000-00000-0000-00000-00000-00000-00000-00000-00000-00000-000000-00000-000000-000000-000000-000000-00000-000000-00000-000000-00000-000000-000000-000000-00 the regulations. Moreover, pile design often uses spring-foundation models based on the lateral force-displacement relationship. [PERSON] et al. [6] presented a series of field lateral load tests based on open-ended steel pipe piles in dense siliceous sand. [PERSON] et al. [7] carried out 12 centrifuge tests to obtain the nonlinear Winkler foundation spring of liquefiable soil. [PERSON] et al. [8] conducted a 1 g model test to obtain the Winkler nonlinear spring of the pile group under the lateral load. Moreover, the nonlinear spring applicable to CFA and Franki piles was proposed [9; 10] based on the test method. As for research on pipelines, based on model tests of buried pipelines, [PERSON] et al. [11] evaluated the applicability of the existing vertical resistance coefficient and proposed modified vertical resistance coefficients for rigid pipelines and flexible pipelines, respectively. Numerical simulation is also an effective method for obtaining the Winkler foundation modulus. [PERSON] et al. [12] improved the foundation coefficients for sticky soil and sandy soil based on the settlement of foundations using ABAQUS 6.14 software. [PERSON] et al. [13] established an empirical calculation formula for the foundation coefficient based on the existing formula for a subgrade modulus and the finite element results. Evidently, the Winkler foundation modulus obtained by the tests is only applicable to a single geological condition, which makes it difficult to deal with different geological conditions in actual engineering. At the same time, the test method used to obtain the Winkler foundation modulus is time-consuming, labor-intensive, and costly. Numerical methods are generally used to correct the modulus of subgrade based on the existing Winkler foundation modulus formula. In this regard, the applicability of numerical methods is also limited. Thus, a theoretical method with wider applications and lower time and cost limitations should be proposed to obtain the Winkler foundation modulus. Based on a theoretical method, [PERSON] [14] obtained the formula of the Winkler foundation modulus by equating the maximum bending moment by the two models, which are an infinite beam on the elastic foundation and an infinite beam on the three-dimensional continuum foundation under a concentrated force, respectively. [PERSON] [15] provided the vertical resistance coefficient of a standard beam with a width of 0.3 m placed on the surface and obtained the modified coefficient for the beam width.