## 3 Test Results and Analysis  
Figure 4 illustrates the relationship between the \(G_{\rm max}\) and \(e\) for the Yangtze River floodplain's soft soils under different \(\sigma_{3\!c}{}^{\prime}\). For the given \(\sigma_{3\!c}{}^{\prime}\) and OCR, \(G_{\rm max}\) decreases as \(e\) increases. This is because the smaller \(e\) is, the looser the soil, the weaker the cohesion and cementation between the particles, the more open the pores in the soil, and the more unstable the soil structure, leading to the smaller \(G_{\rm max}\). In addition, under the same \(\sigma_{3\!c}{}^{\prime}\) and \(e\), \(G_{\rm max}\) improves with increasing OCR, revealing that the \(G_{\rm max}\) is larger, the higher the pre-consolidation pressure. When Figure 4a-d are combined with the given \(e\) and OCR, \(G_{\rm max}\) rises as \(\sigma_{3\!c}{}^{\prime}\) grows. The increase in \(\sigma_{3\!c}{}^{\prime}\) has a noticeable hoop-tightening effect on the soil, which can reduce the void ratio of the soil and strengthen the bond between soil particles. The greater the effective envelope pressure, the greater the soil sample's resistance to shear deformation, as measured by an increase in \(G_{\rm max}\). In each graph, three red trend lines represent the effect of \(e\) on \(G_{\rm max}\), respectively, and [PERSON] et al. [32] proposed the equation \(F(e)\) to characterize the effect of \(e\) on \(G_{\rm max}\). \[F(e)=e^{-d} \tag{3}\]  
where \(d\) is the fitting parameter characterizing the effect of increasing \(e\) on the degree of \(G_{\rm max}\) decay. When \(\sigma_{3\!c}{}^{\prime}\) and OCR are constants, \(d\) can be determined using regression analysis, and the corresponding values of \(d\) are displayed in Figure 5. It reveals that under the same OCR, the increase in \(\sigma_{3\!c}{}^{\prime}\) does not significantly affect \(d\), which remains nearly constant. As OCR rises, \(d\) reduces gradually, with \(d\) values of 2.981, 2.523, and 2.055 for OCR = 1, 2, and 3, respectively. This indicates that the decay rate of \(G_{\rm max}\) with \(e\) is independent of \(\sigma_{3\!c}{}^{\prime}\) but decreases as OCR grows. Figure 6 displays the relationship between the normalized maximum shear modulus \(G_{\max}/F(e)\) and the normalized initial effective confining pressure \(\sigma_{3\mathrm{c}}{{}^{\prime}}/P_{\mathrm{a}}\), where \(P_{\mathrm{a}}\) is the standard atmospheric pressure, assumed to be 100 kPa. For given OCR, \(G_{\max}/F(e)\) grows as \(\sigma 3\mathrm{c}^{\prime}/\mathrm{Pa}\) improves, but the growth rate gradually decreases, and a power function can describe the relationship between \(G_{\max}/F(e)\) and \(\sigma_{3\mathrm{c}}{{}^{\prime}}/P_{\mathrm{a}}\):  
\[G_{\max}=AF(e)(\sigma_{3\mathrm{c}}{{}^{\prime}}/P_{\mathrm{a}})^{n} \tag{4}\]  
where \(A\) is a measured fitting parameter whose value corresponds to the \(G_{\max}/F(e)\) value of the floodplain's soft soil at \(\sigma 3\mathrm{c}^{\prime}=100\) kPa; n is the stress index, describing the influence of \(\sigma_{3\mathrm{c}}{{}^{\prime}}\) on \(G_{\max}/F(e)\). In addition, the \(G_{\max}/F(e)\sim\sigma_{3\mathrm{c}}{{}^{\prime}}/P_{\mathrm{a}}\) relationship curve gradually shift upward as the OCR increases, with the \(A\) value gradually increasing from 31.8 to 42.0 MPa and \(n\) decreasing from 0.552 to 0.465 as the OCR rises from 1 to 3. Figures 5 and 6 demonstrate that the OCR significantly impacts the fitted parameters \(d\), \(A\), and \(n\). In contrast, Figure 7 shows that as OCR enhances, \(d\) grows linearly, \(A\) rises, and \(n\) decreases linearly. The following equations give the relationship between \(d\), \(A\), \(n\), and OCR:  
\[d=0.463\times\mathrm{OCR}+1.594 \tag{5}\]  
Figure 5: Relationship between \(d\) and \(\sigma_{3\mathrm{c}}{{}^{\prime}}\) with different OCR. The \(G_{\max}\) prediction equation that integrates and considers OCR, \(e\), and \({\sigma_{3c}}^{\prime}\) can be established by combining Equations (3) and (4).