Uniaxial cyclic loading

Smooth nonlinear hysteresis model for coupled biaxial soil-pipe interaction in sandy soils

2.4 Numerical verification

2.4.5 Uniaxial cyclic loading

The pipe was pushed cyclically along one direction that was inclined at an angle 𝜃_𝑢 relative to the 𝑥-axis (Fig. 2.11). In this case of loading, loading or unloading takes place concurrently along two directions. The pipe displacement was a 3-cycle sinu- soidal signal, 𝑢 = 𝑢_𝑛sin(2𝜋𝑡/10), where 𝑢_𝑛 is the amplitude, 𝑡 is the time (0–30 s), 𝜃_𝑢 = {0, 30^◦, 45^◦, 60^◦, 90^◦}, and𝑢_𝑛/𝐷 = {0.1,0.3}. The soil experienced weak nonlin- earity when𝑢_𝑛/𝐷 =0.1, whereas it experienced strong nonlinearity when𝑢_𝑛/𝐷 =0.3.

Figs.2.14,2.15,2.16, and2.17present the results of BMBW, FEM, SPH, and the ASCE- recommended bilinear method. Overall, the FEM and SPH results agreed well with each other and jointly showed that the FDCs were indeed smooth, nonlinear curves. The ASCE bilinear method generated straight lines with slopes 𝐾₇₀. The proposed BMBW method

produced true smooth curves which adequately matched those of the numerical simulations.

Finch (1999) conducted a centrifuge experiment showing typical patterns of the FDC for vertical cyclic loading, as shown in Fig. 2.1. The loading, unloading, and reloading phases jointly create a hysteresis loop. The pinching phenomenon only appears during the unloading phase because the gap forms only below the pipe. The FDCs obtained from FEM and SPH in Figs.2.14,2.15,2.16, and2.17clearly depict the hysteresis loops and pinching phenomenon. In those figures, the BMBW method is capable of generating both the loops and pinching zones, in contrast to the ASCE-recommended bilinear method.

-5 0 5

u = 60^°

This study FEM SPH ASCE

-10 -5 0 5 10

u = 45^°

-10 -5 0 5 10

u = 30^°

-10 -5 0 5 10

-6 -4 -2 0 2 4

u = 0^°

Figure 2.14: 𝐹_𝑥-𝑢_𝑥 for small pipe displacement𝑢_𝑛/𝐷 =0.1.

-10 -5 0 5 10

u = 90^°

-10 -5 0 5 10

u = 60^°

-10 -5 0 5 10

u = 45^°

-5 0 5

-6 -4 -2 0 2 4

u = 30^°

This study FEM SPH ASCE

Figure 2.15: 𝐹_𝑦-𝑢_𝑦 for small pipe displacement𝑢_𝑛/𝐷 =0.1.

In general, the pinching is clearer along the 𝑦-axis than along the𝑥-axis. Specifically, the pinching length in the𝐹_𝑦-𝑢_𝑦curves is larger than that in the𝐹_𝑥-𝑢_𝑥curves (Figs.2.14,2.15, 2.16, and 2.17). The reason is obvious: the gap between pipe and soil causing pinching is due to gravity, which acts vertically. On the 𝑥-axis, when 𝜃_𝑢 = 0^◦, there is no pinching,

-20 -10 0 10 20 u = 60^°

This study FEM SPH ASCE

-20 -10 0 10 20

u = 45^°

-20 -10 0 10 20 u = 30^°

-40 -20 0 20 40

-10 -5 0

5 u = 0^°

Figure 2.16: 𝐹_𝑥-𝑢_𝑥 for large pipe displacement𝑢_𝑛/𝐷 =0.3.

-20 0 20

u = 90^°

-20 -10 0 10 20 u = 60^°

-20 -10 0 10 20

u = 45^°

-20 -10 0 10 20

-10 -8 -6 -4 -2 0 2 4

u = 30^°

This study FEM SPH ASCE

Figure 2.17: 𝐹_𝑦-𝑢_𝑦for large pipe displacement𝑢_𝑛/𝐷 =0.3.

as shown in Figs.2.14and2.16. When𝜃_𝑢 =30^◦, 45^◦, and 60^◦, the pinching occurs. The appearance of pinching is attributed to the effect of𝑢_𝑦, demonstrating the coupling effect between vertical and horizontal directions. For both 𝐹_𝑥-𝑢_𝑥, 𝐹_𝑦-𝑢_𝑦, the pinching length increases with displacement𝑢_𝑛and angle𝜃_𝑢. The BMBW results captured this trend well.

Whereas the secant stiffness and ultimate reaction force in the ASCE uncoupled bilinear model remained unchanged, FEM and SPH results showed that they indeed change depend- ing on𝜃_𝑢. Taking𝐹_𝑥-𝑢_𝑥in Fig.2.16as an example, the ultimate reaction forces were about 8, 4, 3, and 2 kN/m when𝜃_𝑢=0^◦, 30^◦, 45^◦, and 60^◦, respectively. The BMBW had similar results as the numerical simulations, indicating that it captures the coupling when the pipe moves obliquely.

Additionally, the coupling effect between two directions is illustrated by drawing the lateral uplift failure envelope, as in Fig. 2.18, in which 𝐹_{𝑢𝑥 𝜃} and 𝐹_{𝑢 𝑦 𝜃} are the ultimate reaction forces along the 𝑥- and 𝑦-axes corresponding to pipe movement angle 𝜃_𝑢. These forces are normalized by the ultimate reaction forces corresponding to purely lateral and uplift

pipe movements, 𝐹_𝑢𝑥0 and 𝐹_{𝑢 𝑦90}. When the inclination angle 𝜃_𝑢 increased, the ultimate normalized horizontal reaction forces decreased and the vertical reaction forces increased, indicating the coupling effect of pipe displacements in the lateral and vertical directions.

The envelope of the proposed BMBW method was close to that of the FEM simulations.

The analytical results ofNyman (1984), and the FEM results ofGuo (2005) andDaiyan (2013) also are plotted in Fig.2.18for comparison.

0 0.2 0.4 0.6 0.8 1

0 0.5 1

BMBW FEM

Nyman (1984) Guo (2005) Daiyan (2013)

Figure 2.18: Lateral uplift failure envelope.

2.4.6 0-Shaped loading

In 0-shaped loading, as shown in Fig.2.19(a), the loading phase occurs in one direction and the unloading phase occurs in another direction. Because the lateral displacement usually is dominant over the vertical displacement in earthquakes, two ratios of displacement amplitude were considered, namely 𝑢_𝑦/𝑢_𝑥 = 0.25 and 0.5. The displacement amplitude along the𝑥-axis is𝑢_𝑛, where𝑢_𝑛/𝐷 =0.1 and 0.3.

-1 0 1

-0.5 0 0.5

0.5 0.25

-1 0 1

-0.5 0 0.5

Figure 2.19: Cyclic displacement loading patterns: (a) 0-shape loading; and (b) 8-shape loading.

Again, in Figs. 2.20 and 2.21, contrary to the oversimplified bilinear ASCE model, the BMBW produced smooth nonlinear curves and captured the dissipated energy via hysteresis loops. The areas of hysteresis loops were narrow for𝑢_𝑛/𝐷 =0.1 and fatter for𝑢_𝑛/𝐷 =0.3, indicating that more energy is dissipated for higher degree of soil nonlinearity.

-10 0 10

-4 -2 0 2 4

u_y/u_x = 0.25

-10 0 10

u_y/u_x = 0.5

-2 0 2

-3 -2 -1 0 1 2 3

u_y/u_x = 0.25

This study FEM SPH ASCE

-5 0 5

u_y/u_x = 0.5

Figure 2.20: 𝐹_𝑥-𝑢_𝑥and𝐹_𝑦-𝑢_𝑦for 0-shape loading and small pipe displacement𝑢_𝑛/𝐷 =0.1.

-20 0 20

-10 -5 0

5 u_y/u_x = 0.25

-20 0 20

u_y/u_x = 0.5

-10 0 10

-6 -4 -2 0 2 4

u_y/u_x = 0.25

This study FEM SPH ASCE

-20 0 20

u_y/u_x = 0.5

Figure 2.21: 𝐹_𝑥-𝑢_𝑥and𝐹_𝑦-𝑢_𝑦for 0-shape loading and large pipe displacement𝑢_𝑛/𝐷 =0.3.

When the pipe was pushed horizontally along branch𝑂 𝐴in Fig.2.19(a), the ASCE method predicted 𝐹_𝑦 = 0, whereas the FEM, SPH, and BMBW results were non-zero values for 𝐹_𝑦 (hysteresis loop branch 𝑂 𝐴 in Figs.2.20 and 2.21). The 𝐹_𝑦 value of the BMBW was different from that of the FEM and SPH models because the simplified assumed function for 𝜒(𝑓_𝜁, 𝜃_{𝑑 𝑢})does not exactly capture the variation of 𝜒in numerical simulations. However, the difference is acceptable considering the simplicity and computational efficiency that the BMBW offers while reproducing the trend of𝐹_𝑦-𝑢_𝑦.

In Figs.2.20 and2.21, as𝑢_𝑦/𝑢_𝑥 increases from 0.25 to 0.5, the corners of𝐹_𝑥-𝑢_𝑥 become rounder, namely the effect of𝑢_𝑦on𝑢_𝑥 becomes increasingly noticeable. In addition, when

𝑢_𝑛 increases, the coupling effect is more pronounced. This is indicated by the rounder corners of𝐹_𝑥-𝑢_𝑥and𝐹_𝑦-𝑢_𝑦 curves in Fig.2.21compared with the corresponding corners in Fig.2.20.

2.4.7 8-Shaped loading

In 8-shaped loading, as displayed in Fig. 2.19(b), both loading and unloading along the 𝑦-axis occur in either loading or unloading along the 𝑥-axis. Figs. 2.22 and 2.23 show results for𝑢_𝑛/𝐷 =0.1, 0.3 and𝑢_𝑦/𝑢_𝑥 =0.25, 0.5. Similar to the case of 0-shaped loading, the hysteresis loops were thinner for𝑢_𝑛/𝐷 =0.1 and fatter for𝑢_𝑛/𝐷 =0.3. This reconfirms that energy dissipation increases with the degree of soil nonlinearity. Moreover, when 𝑢_𝑦 increased, it increasingly affected 𝐹_𝑥-𝑢_𝑥, creating a bulkier hysteresis loop. The FDC predicted by the BMBW model was in good agreement with the results of FEM and SPH.

-10 0 10

-4 -2 0 2 4 u

y/u x = 0.25

-10 0 10

uy/u x = 0.5

-2 0 2

-4 -3 -2 -1 0 1 2 3 u

y/u x = 0.25

This study FEM SPH ASCE

-5 0 5

uy/u x = 0.5

Figure 2.22: 𝐹_𝑥-𝑢_𝑥and𝐹_𝑦-𝑢_𝑦for 8-shape loading and small pipe displacement𝑢_𝑛/𝐷 =0.1.

-20 0 20

-6 -4 -2 0 2 4 6

u_y/u_x = 0.25

-20 0 20

u_y/u_x = 0.5

-10 0 10

-8 -6 -4 -2 0 2 4

u_y/u_x = 0.25

This study FEM SPH ASCE

-20 0 20

u_y/u_x = 0.5

Figure 2.23: 𝐹_𝑥-𝑢_𝑥and𝐹_𝑦-𝑢_𝑦for 8-shape loading and large pipe displacement𝑢_𝑛/𝐷 =0.3.

In Fig.2.23, there are two points at the corners of𝐹_𝑥-𝑢_𝑥, and one point at𝐹_𝑦 =0 of𝐹_𝑦-𝑢_𝑦

where the slope of FDC becomes discontinuous. This occurs because at those points, 𝜁_𝑦 switches between positive and negative values, leading to changes in 𝐹_{𝑢 𝑦} and 𝐾_𝑦, as in Eq. (2.14). Subsequently, this causes discontinuities in 𝜁¤_𝑥 and 𝜁¤_𝑦. This is inevitable when 𝐹_{𝑢 𝑦1}, 𝐾_𝑦1 and 𝐹_{𝑢 𝑦2}, 𝐾_𝑦2 are combined into one single spring. Nevertheless, the discontinuities vanish when 𝐹_{𝑢 𝑦1}/𝐾_𝑦1 = 𝐹_{𝑢 𝑦2}/𝐾_𝑦2. Furthermore, this happens only at isolated points on the FDC and does not affect the general trend of𝐹_𝑥-𝑢_𝑥 and𝐹_𝑦-𝑢_𝑦. 2.4.8 Transient loading

The reaction force of the soil acting on the pipe was investigated for the case of biaxial transient loading. The 1995 Kobe earthquake𝑀_𝑊6.9 ground motions at OSAJ station, 8.5 km from the fault rupture, were used for this purpose (Ancheta et al.,2013). Fig.2.24shows the time histories and pattern of displacement. The lateral displacement was dominant, with a peak value of 88 mm, whereas the peak vertical displacement was 23 mm. The SPH method was chosen for the reference simulations to appropriately model large displacement of the pipe. Fig.2.25compares the response time histories𝐹_𝑥-𝑡 and𝐹_𝑦-𝑡 predicted using the BMBW, SPH, and ASCE bilinear model. The proposed BMBW method captured the responses obtained by SPH to a high degree of accuracy. In contrast, the ASCE model tended to overestimate the 𝐹_𝑥 in regions of large lateral displacement and ignored the coupling effect. Accordingly, 𝐹_𝑦 was considerably different from the values predicted by SPH and BMBW. Fig. 2.26 shows the FDCs 𝐹_𝑥-𝑢_𝑥 and 𝐹_𝑦-𝑢_𝑦 estimated using the aforementioned three methods. Clearly, the shape of FDC from the BMBW matched well that from SPH, with round corners expressing the coupling between lateral and vertical directions.

0 10 20 30 40 50 60

-100 -50 0 50 100

ux

uy

-100 -50 0 50 100

-20 0 20

Figure 2.24: Kobe earthquake signal.

-10 -5 0 5 10

0 10 20 30 40 50 60

-15 -10 -5 0 5

This study SPH ASCE

Figure 2.25: 𝐹_𝑥-𝑡and𝐹_𝑦-𝑡from BMBW, SPH, and ASCE model for Kobe earthquake.