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The static allowable stress field of foundation under strip foundation is constructed by means of stress columns, and the calculation method of the lower bound foundation bearing capacity based on the two-parameter parabolic Mohr yield criterion is proposed. Moreover, the influence of the amount of stress columns and material mechanical parameters on the lower bound bearing capacity is analyzed. The results show that a better solution can be obtained by optimizing the static allowable stress field. However, the improvement of lower bound solution might be inefficient if the stress column amount is large enough. The stresses of the superimposition area show a reduction with the improvement of stress field; on the other hand, the superposed stresses are enhanced ever faster as the involved stress column increases. The tensile-compressive strength ratio has a moderate effect on the lower bound solution. Finally, the reliability of the proposed method is verified by some rock foundation loading tests.

In the mid-20th century, Drucker and Prager combined the static field and the kinematical field and put forward the limit analysis theory including upper and lower bound analysis, which provided a new tool for solving bearing capacity of foundations. In comparison, the lower limit method is not so widely used because it is more difficult to establish a static allowable stress field. Up to now, the lower bound limit method is mainly composed of two implementation means. Firstly, the construction of static permissible stress field is carried out by superposing stress fields or specifying stress discontinuities [

In 1965, Murrell pointed out that there was a power function relationship between the large and small principal stresses of rocks [

In this paper, the parabolic yield criterion with two parameters is investigated based on the nonlinear strength theory and the principle pf “simple expression form, explicit physical meaning, and easy parameter obtainment.” Subsequently, the static allowable stress field conforming to the lower bound theory is constructed by using multiple stress columns in the spatial foundation beneath strip footing. Finally, the lower bound solution of bearing capacity is obtained with the consideration of the nonlinear parabolic strength criterion.

Foundation bearing capacity, Earth pressure, and slope stability are regarded as three classical problems of soil mechanics [_{c}) of the foundation as an elastoplastic material, two methods are usually adopted: one is to investigate the whole evolution process of the foundation, from elastic deformation state to plastic limit state, which is based on the theory of plastoelasticity mechanics (Figure

Deformation curves: (a) common material; (b) ideal elastoplasticity; (c) rigid plasticity.

If there exists a static allowable stress field throughout the whole object without any yield, which can balance with the load acting on the stress boundary, the object will never fail. When the deformation of the object reaches the limit state, the power of the real surface force in the given velocity field is always greater than (or equal to) the power of the corresponding surface force in the same velocity field of any other static allowable field. The lower bound theorem indicates that (1) the ideal object can adjust itself to take on the potential external load; (2) among all the loads corresponding to the static allowable stress field, the ultimate load is the largest.

In general, the lower bound theorem can be cooperated with the upper bound theorem to define the actual interval of the ultimate load. If the upper limit is equivalent to the lower limit, then the ultimate load is a complete solution.

In 1979, the two-parameter parabolic Mohr strength criterion (as shown in Figure

Parabolic Mohr strength criterion in different coordinate systems: (a) _{1}∼_{3}.

Another form of parabolic Mohr criterion was researched which could be stated as [

From equations (

Conclusively, the above two models are completely equivalent to each other. The two-parameter parabolic Mohr criterion is not only the development of the classic Mohr-Coulomb criterion but also the extension of the Griffith criterion, and it has good application prospects. However, there exists an implicit constraint; that is,

In this research, the following assumptions are adopted: (1) the foundation locates beneath a strip footing with a frictionless bottom; (2) the foundation is weightless; (3) the foundation is a rigid-plastic body, subject to the parabolic Mohr yield criterion.

For the simplest case, only one stress column beneath strip footing is taken into consideration (Figure

A single stress column.

Therefore, the lower bound solution of bearing capacity of the foundation (_{z}) could be given by

Subsequently, a static allowable spatial stress field is shown in Figure _{c}.

The stress field constructed by 3 stress columns.

According to [

Rewrite with the spherical stress

While transforming the yield criterion from (

Consequently, the increment of deviator stress Δ

According to the principle of stress superposition, the synthetic hydrostatic stress component is the algebraic sum of the hydrostatic stress of each stress column, and the synthetic deviatoric stress component is the vector sum of the deviatoric stress of each stress column. By combining with Figure

The stress superposition diagram of 3 stress columns.

Apply the cosine theorem in ΔOAB:

In Figure

Combining equations (

The stress status of area ④ in Figure

Consequently, the lower bound solution of bearing capacity of foundation beneath strip footing with 3 stress columns should be

A more sophisticated static allowable stress field composed of 9 stress columns is shown in Figure _{1}, _{2}……, _{9}, which are determined by parabolic Mohr criterion. Also, a horizontal stress column is affiliated in order not to violate the yield law. For the foundation to the left of stress columns, _{1} = _{3} = 0; for the base area under the strip footing, _{1} = _{z} and _{3} = _{x} > 0. According to the nonlinear strength criterion, _{1}>_{2}. The superimposed stress field is shown in Figure

Stress field constructed by 9 stress columns.

Stress superposition diagram of 9 stress columns.

It is clear that

The hydrostatic stresses could be given by

It should be noted that when the material complies with the nonlinear failure criterion, the angle between the velocity vector and the deviatoric stress vector equals _{1} and _{1} will both approach

The solving procedure is shown in Figure

Solving flow.

Figure _{c} = 10 MPa and _{t} = 1 MPa. It can be seen that the lower bound solution of bearing capacity gradually is enhanced and converges as the stress column amount increases, which means a better lower bound solution can be obtained by constructing a more precise static allowable stress field. However, the improvement of lower bound solution might be inefficient and meaningless if the stress column number is large enough. For instance, the lower bound bearing capacities are 56.90 MPa and 57.03 MPa, corresponding to

Correlation between bearing capacity and stress column amount.

Referring to Figure _{101} denotes the hydrostatic stresses of superposition areas within the whole stress field composed of 101 stress columns and _{1001} represents the deviatoric stresses of superposition areas within the whole stress field composed of 1001 stress columns and so on. It shows that although the amount of stress columns changes significantly, the variation regularity of both hydrostatic and deviatoric stress components is quite similar. The stresses of the superimposition area show a reduction with the improvement of stress field; on the other hand, the superposed stresses elevate ever faster with the increase of the stress column amount it contains. Furthermore, the stress state of all superimposition areas does not violate the aforementioned yield criterion.

Stress distribution in superposition areas.

According to equation (_{t}/_{c} once the compressive or tensile strength is specified. Figure _{z}) with _{t}/_{c} = 0.10. Intuitively, the lower bound solution is positively correlated with _{z}∼_{t}/_{c} curve is plotted in Figure _{t}/_{c} grows up from 1 : 25 to 1 : 5, the lower bound bearing capacity decreases by about 30%. Therefore, _{t}/_{c} has a moderate influence on the lower bound bearing capacity.

Correlation between bearing capacity and

Correlation between bearing capacity and _{t}/_{c}.

Comparative study on uniaxial compression and load test of soft rock foundation in Changsha was performed by Peng et al. [_{L} represents the test data of ultimate bearing capacity, _{a} is the characteristic value of foundation bearing capacity, and _{c} stands for the lower bound solution. Interval strengths are adopted since the tensile strength was not clarified in original research, and _{t} is supposed to vary in the range of 0.04_{c}∼0.20_{c}.

Verification of rock foundation loading tests.

Site | _{c} (MPa) | _{L} (MPa) | _{a} (MPa) | _{a}/_{c} | |
---|---|---|---|---|---|

International Finance Square | 2.54 | 13.59 | 2.63 | 1.03 | 12.25∼16.85 |

Provincial Electromechanical Warehouse | 1.99 | 10.70 | 3.50 | 1.76 | 9.59∼13.20 |

Provincial People’s Bank | 1.95 | 11.05 | 3.00 | 1.54 | 9.40∼12.93 |

By adopting a reasonable stress field, the upper or lower bound solution would be very close to the exact result of the problem. Moreover, the reliability of the proposed solution was validated for the consistency of lower bound bearing capacities and the load tests. The errors might be caused by the following: (1) there exists a size effect in the compressive or tensile strength test, the rock sample cannot represent the entire foundation, and the anisotropy of the rock mass, such as joints, cracks, or weathering degree, could also affect loading test results; (2) during load tests, the failure often occurs in the shallow layer with a depth of 1 ∼ 2 times the side length or diameter of the bearing plate; thus, it is difficult to reflect the stress state of deep mass.

Limited by the loading capacity of test devices, it is often difficult to achieve the limit state in load test of rock foundation. Instead, the loading process stops once the load level is no less than 2 times the design requirement of the foundation. Subsequently, 1/3 of the minor value among the proportional limit load and the ultimate load is taken as the characteristic bearing capacity of the rock foundation. Since the theoretical solution is not restricted by the practical working conditions, the characteristic bearing capacity of foundations obtained by this method will not be lower than the test result. Besides, the characteristic value of the bearing capacity of rock foundation can be evaluated by the uniaxial compressive strength of saturated samples in lab (_{rk}) [_{r} is the reduction factor and the recommended values are listed in Table

Recommended values of the reduction factor _{r}.

Integrity of rock foundation | Good | Moderate | Bad |
---|---|---|---|

_{r} | 0.5 | 0.2∼0.5 | 0.1∼0.2 |

Stress columns were utilized to construct the static allowable stress field of the foundation beneath strip footing, and the lower bound solution of the foundation bearing capacity based on the two-parameter parabolic Mohr yield criterion is obtained.

The proposed solution is verified by comparison with the load test, which enriches the research results of lower bound analysis in the field of foundation bearing capacity. Moreover, it provides a reference for the lower bound problems subject to nonlinear failure criterion.

The research shows that the characteristic foundation bearing capacity adopted by practical engineering is too conservative, and the bearing capacity of rock foundations has not been fully considered and utilized.

Part of the raw data cannot be shared at this time as the data also form part of an ongoing study.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work was funded by Natural Science Foundation, China, Grant no. 51408059.