**SUMMARY**

The population in Sweden and around the world is increasing. When population increases, cities become more densely populated and a demand for investments in housing and infrastructure is created. The investments needed are usually large in size and the projects resulting from the investments are often of a complex nature. A major factor responsible for creating the complexity of the projects is the lack of space due to the dense population. The lack of space creates a situation where a very common feature of these types of projects is the use of earth retaining systems.

The design of retaining systems in Europe is performed today based on Eurocode. Eurocode is a newly introduced standard for the design of structures and is developed in order to make it easier to work cross borders by using the same principle of design in all countries. For the design of retaining walls in Sweden, Eurocode uses the old standard as the basis of the design procedure consisting of two separate calculations, ultimate limit state and serviceability limit state. Since soil does not consist of two separate mechanisms consisting of failure and serviceability, this approach to solving engineering problems fails to address the real behavior of soils.

To handle this problem Bolton et. al. (1990a, 1990b, 2004, 2006, 2008, 2010) developed the theory of “mobilized strength design” where a single calculation procedure incorporates both the calculation of deformations and the safety against failure. The calculation uses conservation of energy and the degree of mobilized shear strength to study deformations in and around the retaining system and the safety against failure in mobilizing the maximum shear strength of the soil.

The aim of this thesis was to introduce the theory of mobilized strength design to geotechnical engineers in Sweden working both in academia and in industry. Another aim of the thesis was to develop a tool that could be used to perform calculations of earth retaining systems based on this theory. The development of a working tool has resulted in a Matlab code which can in a simple way be used to calculate both deformations in the retaining system and the safety against failure by using the degree of mobilized shear strength presented in the theory.

The Matlab code can handle ground layering with different shear strengths and weights of the soil. A comparison instrument in a Mathcad calculation sheet have been developed to produce results based on the original theory where the feature of soil layering is not incorporated into the calculation procedure. The thesis shows that the Matlab code developed performs well but is not yet sensitive enough to produce the same results as the Mathcad calculation sheet and needs to be further developed to make it more robust in order to handle all different excavation scenarios.

The theory of mobilized strength design has been introduced to geotechnical engineers in Sweden and the thesis studies the theory and shows the calculation procedure and how the different input values and calculations affect the analysis. The thesis also shows some areas in which the theory and the code can be modified and where further research can be performed in order to make them fully applicable to Swedish conditions. As an example the use of rock dowels drilled into the bedrock and attached to the retaining structure is a common feature for deep excavations in Sweden. Further research can be pursued on how to incorporate the energy stored in the rock dowels into the calculation procedure.

**LITERATURE REVIEW**

The first procedure for designing an earth-retaining structure is an ultimate limit state, or failure load calculation, where forces acting in the soil stratum and on the structural members are investigated. Calculations are performed using partial coefficients on loads and material parameters to make an allowance for a factor of safety against failure. Based on loads and forces acting on the structure, structural members i.e. struts, sheet-piles etc. can be designed.

This chapter has presented different theories to analyze a retaining system for a deep excavation. The first part consists of the theories used in industry today, ULS and SLS, to design earth retaining systems. The second part consists of the theory of mobilized strength design. Based on the information presented, the designer can perform calculations of wall deformations, ground surface settlements and the degree of mobilized shear strength in different layers throughout the soil stratum. The calculations can be compared with results from numerical calculations to see how they compare with each other.

**PRINCIPLE CALCULATION PROCEDURE AND INPUT DATA**

The soil input parameters needed for the analysis is a stress-strain relation and the density from a representative soil sample. The stress-strain relation for soils can be found through different types of laboratory tests. Depending of which zone of the shearing mechanism needed to be simulated; the stress-strain relation can be found through active-and passive triaxial-tests or direct simple shear tests. Since the direct simple shear test is roughly the average of an active- and a passive triaxial-test, Lam (2010) suggests in similarity to O’Rourke (1993) that the results from a direct simple shear test can be used as a base for all zones when designing earth retaining structures for deep excavations.

Evaluation of the mobilized shear strength ratio, β, is recommended to be performed by studying an appropriate stress-strain curve (Bolton et. al. 2004, 2005, 2006, 2007, 2008, 2010, 2011). Mobilized strength design studies the degree of the mobilized shear strength of the soil inside the deformation mechanism. Bolton et. al. defines the mobilization ratio, β, as the ratio between the current level of mobilized shear strength and the peak shear strength, creating a value of the factor varying between zero and one.

**DEVELOPMENT OF A MATLAB CODE **

The stress-strain relation of the soil can be seen in figure 41 and on a logarithmic scale in figure 2.3. In order to make the tables as easy as possible to understand, and use, both tables are presented as strains versus the degree of mobilized shear strength. Figure 42 is presented on a logarithmic scale in order for the small strains to be easily studied and read off. ll curves based on the equation after Vardanega and Bolton (2011) with input values given in table 2, with the exception to the value specified in the figure

**RESULTS**

The same procedure as the one used in the calculations with zones of soils can be used to calculate deformations but in combination with dividing the soil stratum into zones the stratum is also divided into soil layers with equal thickness. The cantilever stage is calculated the same way both in the original theory presented by Bolton et. al. (2004, 2006, 2007, 2008) and in the theory of layered calculations, and thereby the code. The results are therefore exactly the same for the two different procedures.

Again, as for the second excavation stage the total energy from external-and internal work for all zones must agree with each other and the same iterative procedure as for the second excavation stage is performed. This iterative procedure gives a state of equilibrium between external- and internal work when the maximum displacement is assumed to be 6.0 mm.

**DISCUSSION AND FUTURE RESEARCH**

The possibility to analyze the degree of mobilized shear strength in each layer throughout the soil stratum involved in the deformation mechanism is a very powerful tool to study the mechanics of a deep excavation. By studying the degree of mobilization the designer is allowed to make sure that the soil has not gone to failure in any part of the deformation mechanism involved in the analysis. However, the system studies values that are quite small and therefore a high level of accuracy is needed for the analysis to work correctly.

By dividing the soil stratum into soil layers, much greater possibilities for analyzing the results are given. A major contribution to the applicability of the theory and the code is the possibility to estimate the stress-strain curve based on the equation presented by Vardanega and Bolton (2011). The curve is very sensitive to the different input parameters that go into the equation wherefore the designer is given very large opportunities to make good approximations of the shear strength profile in specific cases. The sensitivity of the equation does however also create a possibility of error, which will have a large impact on the results, if correct input values are not used.

**CONCLUSIONS **

The mobilized strength design method is a powerful and simple way to study different parts of the deformation mechanics involved in a deep excavation system. The designer is allowed to study the mechanics of the excavation mechanism that other tools does not. Mobilized strength design can be used as a tool to validate results from finite element analysis of deformations of a retaining structure and the settlements in the soil retained, or as a tool to make starting estimates of the deformations that can be expected from different excavation situations. The theory can also be used to calculate the mobilization factor in the soil involved in the deformation mechanism. By calculating the mobilization factor the designer is allowed to study how far from failure the retained soil is in certain excavation stages and at certain depths in the soil stratum, without having to involve slip surfaces.

The theory itself is not complex and the designer is not required to have a deep understanding of, for example complex theories of soil models in finite element analysis. Instead the theory uses basic principles of soil mechanics to analyze complex geotechnical challenges.

Source: KTH

Author: William Bjureland

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