⚡ electrical engineering

Secant Young’s moduli under total and effective stress conditions for saturated soft clays obtained from tests in the cyclic triaxial chamber

This study investigates saturated soft clays from Mexico City using cyclic triaxial tests to demonstrate that non-uniform pore pressure distribution significantly influences the calculation of secant Young's moduli under effective stress conditions, unlike traditional total stress interpretations, and evaluates the resulting modulus degradation using the González and Romo (2011) constitutive model.

Original authors: Rigoberto Rivera, Carmelino Zea, Osvaldo Flores, Alexandra Ossa, Enrique Gómez, Mary B. Morante

Published 2026-09-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Rigoberto Rivera, Carmelino Zea, Osvaldo Flores, Alexandra Ossa, Enrique Gómez, Mary B. Morante

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Deep beneath the streets of Mexico City, the ground is not solid rock but a thick, water-saturated layer of soft clay. This soil behaves like a sponge that has been squeezed dry and then soaked again; it is heavy, it squishes easily, and it holds a vast amount of water within its tiny spaces. When the earth shakes during an earthquake, this water trapped inside the clay cannot escape quickly. Instead, the shaking forces the water pressure to rise, pushing the soil particles apart and causing the ground to lose its strength, sometimes leading to buildings sinking or tilting. Engineers need to know exactly how stiff this soil is and how it will react to these shaking forces to design safe foundations. To find this out, they usually place a cylinder of the clay in a machine that squeezes it while shaking it back and forth, measuring how much it resists the movement.

For decades, a standard assumption has guided these measurements: that the water pressure inside the clay cylinder is the same everywhere, from the top to the bottom. Because the clay is so thick and slow to let water through, researchers have typically measured the water pressure only at the very bottom of the sample, using that single reading to interpret the soil's behavior in terms of total stress. If the pressure is uniform, the math used to calculate the soil's stiffness is straightforward. However, if the pressure is actually different at the top compared to the bottom, the standard calculation could be misleading, potentially leading engineers to believe the soil is stiffer or weaker than it truly is. This uncertainty matters most in places like Mexico City, where the soil is exceptionally soft and the risk of ground failure during an earthquake is high.

A team of researchers from the National Autonomous University of Mexico set out to test this long-held assumption. They took undisturbed samples of the soft, dark gray clay from the lakebed zone of the Valley of Mexico and placed them inside a specialized machine called a cyclic triaxial chamber. This device acts like a high-tech pressure cooker that can squeeze the soil from all sides while simultaneously pushing it up and down to simulate the rhythmic shaking of an earthquake. The researchers programmed the machine to shake the soil at three different speeds—0.5, 1.0, and 2.0 times per second—mimicking the range of frequencies found in real seismic events. Crucially, they did not rely on a single sensor. Instead, they installed pressure gauges at three distinct locations: at the bottom pedestal, at the top cap, and along the middle side of the clay cylinder. They then subjected the samples to increasing levels of shaking stress, applying twenty cycles of shaking at each step until the soil reached its breaking point.

The results revealed that the old assumption was incorrect. The water pressure generated inside the clay was not uniform; it varied significantly depending on where it was measured. In every test, the pressure was highest at the bottom of the specimen, lower at the top, and almost negligible along the sides. This difference was not a minor fluctuation. As the shaking intensity increased, the gap between the pressure at the bottom and the pressure at the top grew wider. At the highest levels of stress, the pressure difference between the bottom and the top reached 51 kilopascals, while the difference between the bottom and the sides, where values were almost negligible, was as high as 70 kilopascals. The researchers found that shaking the soil faster, at 2.0 times per second, created even larger differences in pressure compared to slower shaking, suggesting that the speed of the earthquake matters just as much as the strength of the shake.

These findings have a direct and dramatic impact on how engineers calculate the stiffness of the soil, a property known as the secant Young's modulus. When the researchers calculated this stiffness using the traditional method—which relies on the pressure measured only at the base and interprets the results in terms of total stress—they got one set of numbers. When they recalculated the stiffness using the actual, measured water pressure at the bottom to determine effective stress, the numbers changed drastically. The soil appeared to be significantly less stiff when the true, high water pressure at the bottom was taken into account. In some cases, the traditional calculation overestimated the soil's stiffness by a factor of more than two compared to the calculation based on the actual bottom pressure. However, when they used the pressure measured at the top of the specimen, the difference was smaller, though still noticeable. The most striking discovery was that because the pressure along the sides was so low, the stiffness calculated using total stress was almost identical to the stiffness calculated using effective stress based on the low side pressure.

The study concludes that for these specific, very soft clays, the standard practice of measuring pressure only at the base is insufficient. The non-uniform distribution of water pressure is a real physical phenomenon that cannot be ignored, especially when the soil is subjected to rapid shaking. If engineers continue to use the traditional method, they risk misjudging how the ground will behave during an earthquake, potentially leading to designs that are either too conservative or dangerously optimistic. The researchers demonstrated that by measuring pressure at multiple points, they could map the true state of the soil and calculate its stiffness more accurately. They also showed that this complex behavior could be described using a mathematical model that fits the experimental data well, allowing for a better understanding of how the soil degrades and loses strength as the shaking continues. This work does not just refine a number; it corrects a fundamental misunderstanding of how water moves and presses within the soft ground that supports some of the world's most vulnerable cities.

Technical Summary: Secant Young's Moduli under Total and Effective Stress Conditions for Saturated Soft Clays

Problem Statement
In geotechnical engineering, the determination of dynamic deformation parameters for saturated soft clays, particularly those of very low permeability like the lacustrine clays of Mexico City, presents significant challenges. A primary issue is the non-uniform distribution of pore water pressure within a soil specimen during undrained triaxial testing. While pore pressure is traditionally measured only at the base of the specimen, literature (e.g., Josseaume, 1971; Kimura and Saitoh, 1983) indicates that in high-plasticity, low-permeability clays, pressure equalization is not immediate. This is exacerbated by the response lag of measurement devices and boundary constraints at the top cap and base pedestal, which create non-uniform strain states. Consequently, the assumption of uniform pore pressure often fails, leading to uncertainties in calculating effective stress parameters. Previous studies on Mexico City clays have typically interpreted dynamic deformation moduli in terms of total stresses or relied on base-only pore pressure measurements, potentially misrepresenting the soil's effective stress behavior under cyclic loading.

Methodology
To address these uncertainties, the authors conducted a series of cyclic triaxial tests on undisturbed, saturated soft clay specimens (high plasticity, CH) obtained from the Valley of Mexico. The testing was performed using an automated, stress-controlled cyclic triaxial apparatus capable of operating at excitation frequencies of 0.5, 1.0, and 2.0 Hz.

Key methodological innovations included:

Multi-point Pore Pressure Measurement: Unlike conventional setups, pore pressures were measured simultaneously at three distinct locations: the base of the specimen, the top cap, and the lateral periphery (mid-height).
Stress Conditions: Specimens were isotropically consolidated to predetermined effective stresses ( $\sigma'_c$ ) and then subjected to stepwise increasing cyclic vertical deviatoric stresses ( $\pm \sigma_z$ ) until failure.
Data Analysis: Hysteresis loops were constructed for both total stresses ( $\pm \sigma_z$ vs. $\pm \varepsilon_z$ ) and effective stresses ( $\pm \sigma'_z$ vs. $\pm \varepsilon_z$ ). The effective stress loops were generated using pore pressure data from the base, top, and periphery to calculate corresponding secant Young's moduli ( $E_z$ and $E'_z$ ).
Modeling: The degradation of the secant Young's moduli with increasing strain levels was fitted using the constitutive model proposed by González and Romo (2011).

Key Results
The experimental data revealed significant non-uniformity in pore pressure distribution and its impact on modulus calculation:

Pore Pressure Distribution: Pore pressures were not uniform. The highest values were consistently recorded at the base, followed by the top cap, while values at the specimen periphery were almost negligible.
- The difference in pore pressure between the base and the periphery ( $2u_b - 2u_p$ ) ranged from approximately 13 kPa at low stress ratios ( $\sigma_z/\sigma_{zf} = 0.25$ ) to 70 kPa at high stress ratios ( $\sigma_z/\sigma_{zf} = 0.90$ ).
- This non-uniformity increased with higher loading frequencies (comparing 0.5 Hz to 2.0 Hz) and higher stress amplitudes.
Impact on Secant Young's Moduli: The non-uniform pore pressure significantly influenced the calculated effective stress moduli ( $E'_z$ $E_{z}^{'}$ ).
- The ratio of total stress modulus to effective stress modulus calculated using base pore pressure ( $E_z / E'_{zb}$ ) ranged from 1.46 to 2.42.
- The ratio using top cap pore pressure ( $E_z / E'_{zh}$ ) was lower, ranging from 1.23 to 1.40.
- Because peripheral pore pressures were negligible, the total stress moduli ( $E_z$ ) and the effective stress moduli calculated at the periphery ( $E'_{zp}$ ) were found to be practically identical.
Frequency and Stress Dependence:
- Total stress moduli ( $E_z$ ) appeared largely independent of the excitation frequency for a given confining stress.
- Effective stress moduli ( $E'_{zb}$ ) showed dependence on excitation frequency, though a clear behavioral law across the tested range (0.5–2.0 Hz) was not established.
- The variation of normalized effective stress moduli ( $E'_z / \sigma'_c$ ) was highly dependent on the consolidation stress level, whereas total stress moduli were not.

Significance and Conclusions
The paper concludes that for saturated soft clays with very low permeability, the traditional practice of measuring pore pressure solely at the specimen base and assuming uniform distribution is insufficient for accurately determining effective stress parameters under cyclic loading. The study demonstrates that:

The non-uniform distribution of pore pressure and strain is a critical phenomenon that must be accounted for when interpreting cyclic triaxial tests on such soils.
Ignoring this non-uniformity leads to significant discrepancies between total and effective stress moduli, with the magnitude of the error depending on the location of the pore pressure measurement used for calculation.
While the degradation curves of secant Young's moduli can be established for both total and effective stresses using analytical models (such as González and Romo, 2011), the choice of pore pressure data (base vs. top vs. average) fundamentally alters the resulting effective stress parameters.

The authors assert that these findings are particularly relevant for seismic analysis in Mexico City, where the tested frequency range (0.5–2.0 Hz) is representative of local earthquake characteristics. The research provides a more rigorous basis for interpreting dynamic soil behavior by highlighting the necessity of multi-point pore pressure monitoring to resolve the uncertainties inherent in effective stress calculations for low-permeability clays.

More like this