It is of great significance to study the interactions between structures and supporting soils for both structural engineering and geotechnical engineering. In this paper, based on the refined two-parameter elastic foundation model, the bending problem for a finite-length beam on Gibson elastic soil is solved. The effects of axial force and soil heterogeneity on the bending behaviours and stress states of beams on elastic foundations are discussed, and the parameters of the physical model are determined reasonably. The beam and elastic foundation are treated as a single system, and the complete potential energy is obtained. Based on the principle of minimum potential energy, the governing differential equations for the beam bearing axial force on the Gibson foundation are derived, and the equations for attenuation parameters are also defined. The problem of the unknown parameters in foundation models being difficult to determine is solved by an iterative method. The results demonstrate that this calculation method is feasible and accurate, and that the applied theory is universal for the analysis of interactions between beams and elastic foundations. Both axial force and soil heterogeneity have a certain effect on the deformation and internal force of beams on elastic foundations, and the vertical elastic coefficient of foundations is mainly determined by the stiffness of the surface soil. Additionally, attenuation parameters can be obtained relatively accurately by an iterative method, and then the vertical elastic coefficient and shear coefficient can be further obtained. This research lays a foundation for the popularisation and application of the two-parameter elastic foundation model.
With the increasing number of construction projects worldwide, many beam, plate, and shell structures have been developed, and strict standards and requirements have been proposed for structures and foundations. Foundation engineering is concealed by superstructures, which has a significant impact on the safety and economy of structures and foundations. Various quality-related accidents occur in the engineering fields, which not only cause huge losses but also make it difficult to take appropriate measures for strengthening structures. Elastic foundation beams have been widely used in structural engineering. They are the basic components in structures such as roads, bridges, and high-rise building foundations. It is critical to apply structural components in engineering practice reasonably and safely, and these concerns have attracted the attention of various academic and engineering circles [
The comprehensive analysis and detailed study of classical components have become important and difficult problems. Many scholars have focused on studying the interactions between structures and complex media and have obtained various theoretical systems for describing structures resting on foundations. Kimençe and Ergüven [
Many scholars have studied beams on two-parameter foundations using the direct method, Galerkin method, power series method, differential operator series method, and finite element method. However, on the one hand, the influence of axial force on foundation beams has not been considered in previous studies. In practical engineering, foundation beams are often subjected to axial forces, so it is necessary to study the influence of such forces. Additionally, designers often simplify the foundation beams to have free ends. However, in real-world scenarios, boundary conditions are not always free, and clamped support is often applied at both ends of a beam. Existing research on this type of problem is very rare.
On the other hand, the traditional Vlasov foundation model considers an elastic layer as a homogeneous and isotropic body. However, the elastic modulus of soil is considered as a linear variable along the depth of a foundation, which more closely matches its real-world behaviour. Dempsey and Li [
Finally, the two-parameter foundation model has the advantages of simple numerical treatment and perfect theory, where two independent elastic parameters are used to represent the characteristics of foundation soils. However, the traditional Vlasov model requires the estimation of an attenuation parameter
Regarding the foundation soil, the inhomogeneity of Gibson soil is more in line with the values in engineering practice, compared to considering the simple homogeneous and isotropic soil. In this study, finite-length beams on refined Vlasov foundations are analysed based on Gibson characteristics. The interactions between foundations and finite-length beams have been investigated systematically, and the influences of soil inhomogeneity and axial force on the bending of beams on elastic foundations have also been examined. According to the principle of energy variation, the governing equations for finite-length beams on Gibson two-parameter elastic foundations are established. Additionally, the equations that the attenuation parameters must satisfy are obtained based on variation. Finally, the attenuation parameters obtained using an iterative process are used to calculate the two characteristic parameters of the mathematical model, and relatively accurate deflections and internal forces of beams are obtained.
The novelty and importance of this paper include (1) the refined Vlasov foundation model is adopted to simulate the mechanical behaviour of the Gibson soil. (2) Using the principle of minimum potential energy, the governing equations and boundary conditions for finite-length beams resting on the refined Vlasov elastic foundations are derived. (3) Using an iterative process, the consistent values of characteristics parameters are obtained. (4) A comparative analysis between the refined Vlasov foundation and the traditional Vlasov foundation model is also carried out. (5) The effects of axial force and Gibson soil heterogeneity on the bending behaviours and stress states of beams resting on the refined Vlasov elastic foundations are discussed.
As shown in Figure
Finite-length beam on a Gibson two-parameter elastic foundation.
In this paper, soil inhomogeneity is considered. It is assumed that the foundation is a Gibson soil whose elasticity modulus changes linearly with depth. The elasticity modulus at the top and bottom of the foundation is denoted as
The elasticity modulus
One can see that the soft and hard conditions of the foundation soil depend on the value of the dimensionless parameter
The total potential energy of the beam-foundation system is
When the Gibson foundation is deformed, the horizontal displacement component
The deformation potential energy of the Gibson foundation can be rewritten as
The complete potential energy of the system is calculated as follows:
By minimising the function
The governing differential equations for the beam can be derived through complex variational deduction.
The foundation soil is divided into the part under the beam and the part outside the beam. The governing equations for the foundation soil outside the beam can be obtained as follows.
Left side of the beam:
Right side of the beam:
The attenuation expressions of the vertical displacement components on the surface of the foundation soil outside the beam are defined as follows:
Similarly, by collecting the coefficients of
Additionally, the boundary conditions of the beam with axial force on the Gibson elastic foundation can also be obtained from formula (
If the beam is fixed at one end and clamped at the other one end, then
If the beam is simply supported at both ends, then
If the two ends of the beam are free, then
The following two types of boundary conditions for a finite-length beam on an elastic foundation are discussed: (1) the two ends of the beam are free and (2) the beam is fixed at one end and clamped at the other end.
First, if both ends of a finite-length beam are free and it is subjected to a uniformly distributed load
Second, if a finite-length beam is fixed at one end and clamped at the other end, and it is subjected to a distributed load
The uniformly distributed load
The concentrated load
To compare coefficients, it is necessary to expand
The equations above are substituted into the relevant governing differential equations and boundary conditions, and the algebraic equations for the two boundary conditions can be obtained by comparing the coefficients. The number of equations is the same as the number of undetermined coefficients, so the problem can be solved.
An effective iterative technique for solving the problem of a finite-length beam on a Gibson elastic foundation using the refined Vlasov model is detailed as follows:
Principles of solid mechanics are used instead of an experimental or empirical evaluation of the attenuation parameter
The refined Gibson foundation model is simplified to the traditional Vlasov elastic foundation. The parameters are the length of the beam
When the axial force
The bending problem of beams with axial forces on a Gibson two-parameter elastic foundation is considered. The parameters to be calculated are the length of the beam
The calculation results for beams with free ends are listed in Table
When
Further analysis reveals that the inhomogeneity of Gibson foundation soil has a significant influence on the bending behaviour of the elastic layer around the load. The bending deflection of a finite-length beam on a Gibson elastic foundation is mainly affected by the rigidity of the surface soil and less affected by the deep portions of the foundation.
Calculation results for beams with free ends on a Gibson elastic foundation in Example
Parameters | ||||
---|---|---|---|---|
½ | 0.62105 | 1.1899 | 0.8191 | 4.185 |
1 | 0.62106 | 1.6204 | 1.3171 | 3.113 |
2 | 0.62106 | 2.4812 | 2.3133 | 2.108 |
Calculation results for beams with one end fixed and the other end camped on a Gibson foundation in Example
Parameters | ||||
---|---|---|---|---|
1/2 | 0.6211 | 1.1896 | 0.8182 | 2.932 |
1 | 0.6211 | 1.6205 | 1.3159 | 2.399 |
2 | 0.6211 | 2.4824 | 2.3113 | 1.790 |
Deflection of a beam on a Gibson two-parameter foundation.
Rotation angle of a beam on a Gibson two-parameter foundation.
Bending moment of a beam on a Gibson two-parameter foundation.
Shear force of a beam on a Gibson two-parameter foundation.
Subgrade reaction of a Gibson two-parameter foundation.
In Table
Attenuation parameters for two-parameter foundations in the examples.
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
1.0000 | 0.6993 | 0.5489 | 0.4737 | 0.4361 | 0.4173 | 0.4079 | 0.4032 | 0.4009 | 0.3997 | 0.3991 | 0.3985 | |
1.0000 | 0.8105 | 0.7158 | 0.6684 | 0.6447 | 0.6329 | 0.6270 | 0.6240 | 0.6225 | 0.6218 | 0.6211 | ||
1.0000 | 0.8110 | 0.7160 | 0.6680 | 0.6450 | 0.6330 | 0.6270 | 0.6211 |
By observing the deformations and internal forces of the foundation beams under different axial forces (see Figures
Deflection of finite-length beams with different axial forces.
Rotation angle of finite-length beams with different axial forces.
Bending moment of finite-length beams with different axial forces.
Shear force of finite-length beams with different axial forces.
It can be seen from Figures
Vertical elastic coefficient
Shear coefficient
Attenuation parameter
Deflection
The study also reveals that the harder the surface foundation is, the greater the vertical elastic coefficient
It can be seen from Figures
Vertical elastic coefficient
Shear coefficient
Attenuation parameter
Deflection of finite-length beams for different foundation depths.
The further study also reveals that, with an increase in the elastic modulus of a beam, the deflection of a finite-length beam on elastic foundations decreases. However, the values of the attenuation parameter
Based on the refined Vlasov elastic foundation model, this paper analyses and solves the bending problem of finite-length beams on elastic Gibson soil foundations. The influences of soil heterogeneity and axial force on beam bending behaviours and foundation characteristic parameters are also investigated. Some conclusions can be drawn based on the numerical results. The work begins with total deformation potential energy of a foundation-beam system. Through theoretical derivation, it is found that there are no differences between the equations for beams on Gibson foundation and those on traditional two-parameter foundation. However, the relevant model parameters change. If the soil of the refined Vlasov foundation was assumed to be a classical homogeneous medium, the results in this study would be degraded to the classical case of finite-length beams on classical Vlasov elastic foundation. Examples are given to demonstrate the practical application of the refined foundation model. The presence of axial force makes the midspan deflection, maximum bending moment, and rotation angle of a foundation beam increase. Axial force has some influences on the shear force of a beam. However, the degree of influence varies with the position on the beam. It is feasible to ignore the influence of axial force on the peak shear force in engineering calculations. However, the influence of axial force on shear force cannot be ignored at the positions near the two beam ends. The Gibson soils have a certain effect on the deflection, internal force, and various characteristic parameters, which should be considered in practice. The results show that the mechanical behaviour of a foundation is mainly determined by the characteristics of the shallow foundation soil under the superstructure, rather than the deep parts of the foundation. In engineering practice, to improve the performance of structural foundations, we can consider enhancing the performance of the shallow portions of foundations. Considering the properties of Gibson soils, the refined Vlasov foundation model still uses two independent parameters to express the compression and shear properties. For the key attenuation parameter, an iterative method can yield superior results. The characteristics of foundation models do not depend on a certain parameter of soils and structures, but are related to multiple coexisting physical quantities. This indicates that the attenuation parameters derived from experiences or experiments are not accurate and reliable. This paper enriches and expands the content of the Vlasov model and promotes its widespread application.
All data contained in this study are available upon request from the corresponding author.
The author declares no conflicts of interest.