A machine approach to predicting pressure response in sands

ABSTRACT Shear stress history controls the pore pressure response in liquefiable soils. The excess pore pressure does not increase under cyclic loading when shear stress amplitude is lower than the peak prior amplitude -- the shielding effect. Many sophisticated constitutive models fail to capture the shielding effect observed in the cyclic liquefaction experiments. We develop a datadriven machine learning model based on the LSTM neural network to capture the liquefaction response of soils under cyclic loading. The LSTM model is trained on 12 laboratory cyclic simple shear tests on Nevada sand in loose and dense conditions subjected to different cyclic simple shear loading conditions. The LSTM model features include the relative density of soil and the previous stress history to predict the pore water pressure response. The LSTM model successfully replicates the pore pressure response for three cyclic simple test results considering the shielding and density effects.

 

During an earthquake, the pore pressure response of saturated soils controls their behavior and is related to liquefaction triggering. Liquefaction occurs when excess pore water pressure exceeds effective stress, resulting in a significant loss of strength and stiffness. Cyclic simple shear tests on saturated Nevada sand under different densities and input stresses (Sideras, 2019) show that the stress history plays a vital role in the generation of pore water pressure. When the soil experiences a shear stress pulse amplitude larger than the prior peak amplitude of the stress history, it causes a rapid increment in pore pressure. On the other hand, when the stress amplitude is lower than the prior peak amplitude, the stress does not contribute to any pore pressure generation unless the soil reaches the phase transformation state. This stress history-dependent response of pore water pressure is called the shielding effect. Ishibashi et al.

 

(1977) proposed a relationship to predict pore water pressure build-up of saturated sandy soil with loading cycles based on the number of cycles, shear stress at the previous stress, and pore water pressure at the previous cycle. This predictive model can account – 2 – for the discrete pore water pressure value after each cycle of harmonic loading but not the continuous build-up of stresses with time. Because earthquake loadings are irregular, not harmonic, and the pore water pressure responds to the continuous stress history for every time step, not after each cycle of previous stress, we should investigate the behavior in a continuous framework. Sideras (2019) investigated if numerical models could capture the continuous response of pore water pressure, including the shielding effect (Fig. 1). Sideras used the Pressure

Dependent Multi-yield 02 (PDMY02) (Yang et al., 2008) and the PM4Sand (Boulanger and Ziotopoulou, 2017) model to simulate single-element cyclic simple shear tests on OpenSees and FLAC. Both PDMY and PM4Sand models could not capture the shielding effect observed in the laboratory experiments.

 

Fig. 1 shows the pore pressure response for cyclic simple shear test on Nevada sand. The pore water pressure 𝑟𝑢 is the ratio between excess pore water pressure and effective stress. Laboratory tests and constitutive models predict the increase in the pore pressure ratio 𝑟𝑢 until around 36 seconds as shear stress (𝜏) amplitude grows (region A). At 36 seconds, the sample experiences a large stress pulse, and the subsequent stress does not exceed its amplitude until 57 seconds. As a result, 𝑟𝑢 in test data maintains a nearly constant level. However, both constitutive models show a clear increase in 𝑟𝑢 during this period (region B). From 57 to 60 seconds, 𝑟𝑢 begins to increase again in the test data because the loading amplitudes exceed the prior one (region C). The constitutive models also show these increases but to a lower extent. Region D again experiences the shielding effect; unlike previous times, the models capture the shielding effect in this region. In addition, the model predictions of time to liquefaction (when 𝑟𝑢 reaches 1.0) slightly deviate compared to the test data. The constitutive models also do not reproduce the fluctuations in the pore pressure response. Figure 1. Pore pressure response 𝒓𝒖 to cyclic simple shear test using PDMY02 and PM4Sand models (Siders, 2019), where 𝛕 is the input shear stress. A B C D – 3 – Recently, there have been efforts in the geotechnical field to describe soil behaviors using a data-driven approach. Given a sufficient amount of data on the behavior of pore water pressure responding to various input stresses, one can use machine learning and neural networks to find a model that best explains such behavior. We use a long short-term memory (LSTM) neural network (Hochreiter and Schmidhuber, 1997) trained on cyclic simple shear test data from (Kwan et al., 2017) to capture the pore pressure response. METHOD Long short-term memory neural network. Long Short-Term Memory (LSTM) network is a variant of the recurrent neural network (RNN). RNN can process and predict sequential data and is capable of remembering patterns observed in the input sequence. However, RNN fails to model long sequences as it suffers from vanishing and exploding gradient problems (Bengio et al., 1994). LSTM solves this problem by introducing forget, input, and output gates (Fig. 2).

 

These gates, which are a 0-1 valued matrix, regulate what information from the previous sequence and current feature to consider or forget, and which information to pass to the next cell. The previous information is represented as the previous cell state (𝐶𝑡−1) and hidden output (ℎ𝑡−1). The current feature is 𝑥𝑡 . The hidden output of the current cell is ℎ𝑡 and the current cell state is 𝐶𝑡 . Figure 2. LSTM cell architecture. The subscript 𝒕 stands for the time step. 𝒙, 𝒉, and 𝒄 are the input, hidden output, and cell state. 𝒇𝒕 , 𝒊𝒕 , and 𝒐𝒕 are the forget, input, and output gate at 𝒕. 𝝈 and 𝒕𝒂𝒏𝒉 are the sigmoid and hyperbolic tangent function. ⊙ and ⊕ are the element-wise product and summation. The details of the LSTM cell are as follows. Forget gate (𝑓𝑡 ) regulates how much information of the previous cell state (𝐶𝑡−1) to bring into the current cell state by referring to the previous hidden output (ℎ𝑡−1) and current feature (𝑥𝑡 ) (Eq. 1). 𝐶𝑡−1 is multiplied by 𝑓𝑡 (Eq. 5) squeezing values of elements in 𝐶𝑡−1 to be smaller or make it zero if they should not be included in the current cell state. The other gates do a similar operation. Input gate (𝑖𝑡 ) regulates the extent of candidates (𝐶𝑡 ̃ ) (Eq. 3) of the new information (𝑥𝑡 ) to be included in the current cell state (Eq. 2). The result of the input gate is added to the cell state determined in the forget gate to make the – 4 – final current cell state (𝐶𝑡 ) (Eq. 5), which will be passed to the next cell. Output gate (𝑜𝑡 ) regulates which output to transfer to the next cell (ℎ𝑡 ) based on 𝐶𝑡 , 𝑥𝑡 , and ℎ𝑡−1 (Eq. 4).