Chapter 5 Effects of Cross Walls on Wall Displacement
5.3 Numerical Results and Comparisons
5.3.1 Results of numerical analyses
Summarizing the numerical results, the ratio of the maximum wall displacement divided by the excavation depth (𝛿ℎ𝑚/𝐻𝑒) versus the spacing of cross wall (𝑑𝑐) are shown in Figure 5.2 to Figure 5.5 and these figures respectively represent a specific site dimension with different soil strength. For example in Figure 5.2, the numerical results are for 60 m by 60 m site with different soil strength. It can be seen that the ratio of the maximum wall displacement divided by the excavation depth reduces significantly as the spacing is decreased from 60 m to 20 m. Especially for the weaker soil, the reduction rate is more obvious. In addition, these results can be redrawn with site dimension as variable, the results are shown in Figure 5.6 to Figure 5.11.
All numerical results can be summarized in a single chart to show the effect of the spacing, which is shown in Figure 5.12. It is observed that the wall displacement decreases linearly with the spacing of cross walls, and it might be able to draw a straight line to describe the relationship between the spacing and the wall displacement ratio. In other words, there is a unique relationship between the wall displacement ratio and its corresponding spacing of cross walls. It is also observed that a same spacing of cross walls would result in a similar displacement of diaphragm wall regardless the size of these excavation zones. For instance, for strength ratio of 0.18, the 30 m spacing of Square60
and Square30 models had similar maximum wall displacement of about 0.58% and 0.56%
of the excavation depth, respectively. In addition, for the same soil strength, the 10 m spacing of the Square60, Square30 and Square20 models did not have the similar maximum wall displacement though the maximum wall displacement ratio is below 0.2%.
A larger excavation zone will result in a larger maximum wall displacement. This can be attributed to the cross walls above the excavation surface are removed by step-by-step excavation. In addition, the maximum wall displacement would usually happen at the center section of diaphragm wall or the middle of the two adjacent cross walls, so that the possible location of the maximum wall displacement would be more further away from the corner of the diaphragm wall for the larger excavation zone. The site dimension of Square60 is larger than those of Square30 and Square20, resulting in a larger maximum wall displacement despite the spacing of cross walls are the same for sites with different size.
Linear relationship between the spacing of cross walls and wall displacement ratio are presented in Figure 5.13 for various soil strength. Lines shown in Figure 5.13 are for the undrained shear strength ratio equals to 0.18, 0.22, 0.24, 0.27, 0.30 and 0.33, respectively. The slope and the intercept of these lines depend on the undrained shear strength ratio. It is interesting to note that the slope and the intercept of those lines could be represented by a simple equation in exponential form as shown in the following.
𝐴 = 0.0011 × 𝑚−1.87 𝐵 = −0.0001 × 𝑚−4.4
𝛿ℎ𝑚⁄𝐻𝑒(%) = (0.0011 × 𝑚−1.87) ∙ 𝑥 + (−0.0001 × 𝑚−4.4)
where 𝐴 and 𝐵 are the slope and the intercept of the lines, respectively; 𝑚 is the undrained shear strength ratio; 𝑥 is the spacing of cross walls.
In Figure 5.13, it is noted that these lines seems to have a limited cut-off line at about 15 m, which is the spacing of cross walls that develops the best efficiency in limiting wall displacement. Once reached the cut-off line, decreasing the spacing further has very limited effect on reducing the displacement of diaphragm wall. An optimal spacing of 15 m is in agreement with current design practice that the best spacing of cross walls should be around 15 m.
Using undrained shear strength ratio of 0.18 and 0.33 as the lower and upper boundaries of soil strength, the range of possible wall displacement ratio can be defined in Figure 5.14. The boundaries of displacement ratios can be applied in other cases to verify its applicability, which is discussed in the next chapter.
5.3.2 Comparing the numerical results with predictions by the regression equation
The numerical results are further discussed in this section. As shown in Appendix A, the system stiffness (𝑆), combined system stiffness (𝑆𝑐), factor of safety against basal heave (𝐹𝑏) and adjusted factor of safety against basal heave (𝐹𝑏_𝑎𝑑𝑗) are calculated for all numerical results. The combined system stiffness and adjusted factor of safety against basal heave can further be used in conjunction with the regression equation to predict the corresponding wall displacements ( 𝛿𝑟𝑒𝑣 ) under the influence of cross walls. If the regression equation is valid to certain extent, the predictions (𝛿𝑟𝑒𝑣) by regression equation should be close to the numerical results (𝛿3𝐷).
Unfortunately, the comparison shows the opposite. It is found that most of the predictions by regression equation are far smaller than the numerical results. In addition, some of the adjusted factor of safety against basal heave are unreasonably high as the small zones divided by cross walls may have an extremely high length to width ratio. For instance, the adjusted factor of safety against basal heave is 11.21 for case “S60, m033, 5h0v”, which is an unreasonably high value induced by a length to width ratio of 6 as shown in Table 5.6 extracted from Appendix A. The high length to width ratio results in an overestimation of the cross wall effect.
More often than not, the full length of cross walls is used in estimating its
cross walls. There is doubt that if the full length of cross walls should be used in estimating the associated strengthening effect. From a theoretical point of view, the development of basal heave failure surface is subdued by the friction between cross walls and soil, this is a concept also shared by Hsieh and Lu (1999). Since the potential basal heave failure surface is limited by the dimension of the project site, perhaps an equivalent length (𝐿𝑒) of cross walls should be used instead of the full length of cross walls. In this study, 𝐿𝑒 is taken as the radius of the potential failure surface, which in turn is a function depending on the width of the excavation zone or the distance between the excavation surface and the stiff soil layer as proposed by Terzaghi (1943). As shown in Appendix B, it appears that an equivalent length (𝐿𝑒) should be used in estimating the effect of cross wall. Replacing the full length of cross wall with the equivalent length seems be better to calibrate the adjusted factors of safety against basal heave. Instead of an excessive value of 11.21, the maximum adjusted factor of safety against basal heave is now 4.3 as shown in Table 5.7 extracted from Appendix B.
It is also noted that the applicability of regression equation has its limit. First, if the cross walls spacing is larger than 30 m or the undrained shear strength ratio is higher than 0.30, the predictions by regression equation is not very reasonable. Second, another interesting aspect to note is the adjusted factor of safety against basal heave will reach a peak value as the spacing of cross walls assumes a value of 20 m. As the spacing of cross
wall is reduced or the number of cross walls increased, the adjusted factor of safety against basal heave is found to be lower as shown in Table 5.7. This observation is against basic cognitions that the adjusted factor of safety against basal heave should increase with the number of cross walls. One possible explanation for this phenomenon is that the adjusted factor of safety against basal heave is a function of site dimension. As the spacing of cross walls reduced, the depth of potential failure surface is limited to a shallower depth and the undrained shear strength ratio is smaller, resulting in an adjusted factor of safety against basal heave that has a smaller value than expected.
In summary, as the spacing of cross walls approaches 15 m, the effect will reach its peak value. Further reducing the spacing between cross walls will not effectively result in further reduction on wall deflection. Therefore, the effects of cross walls on retaining wall would not develop infinitely to an extremely low level. It appears that there is an ultimate low value of wall displacement even if the spacing of cross walls is reduced to a very small number. Hence, the regression equation is most suitable for excavations with cross wall spacing less than 15 m and the undrained shear strength ratio less than 0.30.
Table 5.1 Domain size and spacing of cross walls
Table 5.2 Assumed excavation sequence of the parametric studies
Phase Type depth Remark
Table 5.3 Structural parameters of the parametric studies
Table 5.4 Strut parameters of the parametric studies
Type Depth E A EA EA Preload
Unit m kgf/cm2 cm2 kgf kN kN
H400 × 400 × 13 × 21 - 2.04E+06 218.7 4.46E+08 4.38E+06 490
Table 5.5 Assumed soil parameters of the parametric studies
Type depth to γs Eu su,ref νu Eu,inc su,inc zref Rint
Table 5.6 The parameters and predictions for S60/m033 models by using the full length of cross walls
Table 5.7 The parameters and predictions for S60/m033 models by using the equivalent length of cross walls Type Spacing y
Figure 5.1 Plan layout of the cross walls
Figure 5.2 δhm/He versus cross wall spacing for Square 60
Figure 5.3 δhm/He versus cross wall spacing for Square 42
Figure 5.4 δhm/He versus cross wall spacing for Square 30
Figure 5.5 δhm/He versus cross wall spacing for Square 20
Figure 5.6 δhm/He versus cross wall spacing (m=0.33)
Figure 5.7 δhm/He versus cross wall spacing (m=0.30)
Figure 5.8 δhm/He versus cross wall spacing (m=0.27)
Figure 5.9 δhm/He versus cross wall spacing (m=0.24)
Figure 5.10 δhm/He versus cross wall spacing (m=0.22)
Figure 5.11 δhm/He versus cross wall spacing (m=0.18)
Figure 5.13 Linear relationship between displacement ratio and spacing of cross walls