Prediction of time of liquefaction using kinetic and strain energy

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Abstract

The time of liquefaction triggering during a strong ground motion can have a large influence on the expected level of foundation and superstructure damage. To enable simple, yet accurate estimates of the triggering time, the build-up of pore pressure needs to be understood in terms of cumulative measures of ground motion intensity. This paper develops a theoretical framework and simple procedure to predict the build-up of excess pore pressure based on the principles of conservation of energy. The liquefaction resistance is first quantified in terms of cumulative absolute change in strain energy, which is shown through the evaluation of experimental cyclic simple shear tests to be insensitive to loading amplitude. A ground motion intensity measure is presented that uniquely calculates the cumulative absolute change in kinetic energy. This intensity measure is then used to provide an exact analytical solution for the cumulative absolute change in strain energy at any depth in a homogenous linear elastic soil deposit using the novel, nodal surface energy spectrum (NSES). A simple reduction to the NSES is proposed for viscous and nonlinear soil deposits, as well as a correction for changes in stiffness between layers of soil. The estimation of strain energy and build-up of pore pressure using the simple NSES method was applied to 500 randomly generated soil deposits using a range of different ground motions and validated against nonlinear total stress and nonlinear effective stress time-history analyses, with the NSES method providing a high level of accuracy. The proposed spectrum based solution provides an efficient and physically consistent procedure for the prediction of excess pore pressure build-up.

Introduction

The generation of excess pore water pressure and liquefaction can dramatically change the dynamic response of a soil deposit and interacting structures. Thus the time at which liquefaction occurs, may have a significant influence on the performance of a structure during a seismic event.

Recent work by Wotherspoon et al. [1]; showed that the expected level of surface shaking and seismic energy entering the building is strongly conditioned on the time of liquefaction. In fact Kramer et al. [2], demonstrated that the strong change in frequency content and amplitude of the surface acceleration due to liquefaction can be used to detect the occurrence of liquefaction. Recent centrifuge tests by Jafarian et al. [3] have shown a clear change in the rate of settlement and surface acceleration with the triggering of liquefaction. Bird et al. [4] recognised that damage can be generated through liquefaction-induced effects and ground shaking, and proposed a utility function to add the two causes based on the time at which liquefaction occurred. Kramer et al. [5] proposed a framework for assessing liquefaction effects (e.g. lateral spreading) based on ground motion intensity measures computed before and after the triggering of liquefaction. And recently Bouckovalas et al. [6], proposed a method for obtaining the surface shaking response spectra using the equivalent linear analyses of liquefied and non-liquefied deposits and taking the response from the pre and post liquefaction segments of the ground motion.

There are advanced nonlinear effective stress analysis techniques for evaluating the time of liquefaction. Unfortunately, these approaches require an extensive number of soil parameters, and non-trivial decisions about constraining the domain of the analysis (e.g. depth of the model). Whereas, simplified stress-based methods that quantify the soil liquefaction capacity in terms of the cyclic resistance ratio (CRR) (the amplitude of cyclic stress divided by the initial at rest vertical effective stress required to cause liquefaction under a certain number of cycles of equal stress amplitude), have been developed only for the assessment of liquefaction triggering, and often have biases or simplifications, that present significant drawbacks when used beyond their initial purpose (see Refs. [7,8]). The direct application of equivalent cycle counting methods (e.g. Ref. [9]), are considered to overcome some of the issues with estimating the time dependent cyclic demand but require the full stress time series and at least two parameters to define the CRR versus number of cycles relationship. Additionally, the typically application of these procedures applies the Palmer-Miner cumulative damage hypothesis which assumes a high number of cycles with essentially elastic behaviour, whereas liquefaction inherently involves a large change from initial elastic soil deformation behaviour, and therefore additional corrections should be applied to determine equivalent cyclic loading [10]. Another possibility is to correlate liquefaction triggering with the cyclic strain amplitude (e.g. Ref. [11], as shear strain is strongly correlated to volumetric change [12]. While strain-based approaches can potentially be less influenced by soil fabric, ageing and confining stress, they still suffer from the same issues of the stress based methods, related to estimating the demand from a seismic wave. Alternatively, the dissipated energy of the soil during loading is closely linked to soil grain movement [13] and has been shown experimentally to be a loading independent measure of the liquefaction resistance of the soil (e.g. Refs. [[14], [15], [16], [17], [18], [19]]). However, methods that adopt dissipated energy have two major drawbacks, one is that the estimation of the dissipated energy within a soil profile from a seismic shear wave is far from trivial, and very dependent on soil characteristics and shear strain [20]. Secondly, the dissipated energy rapidly increases as the soil approaches liquefaction (in tests with equal cycles of stress amplitude), and therefore a small change in the criteria for liquefaction triggering (e.g. change the limiting of excess pore pressure ratio from 0.95 to 0.98), can have a large impact on the evaluated capacity, though adjustments can be made to compute a dissipated energy under equivalent ‘total stress’ conditions that removes this effect [10].

To overcome some of the drawbacks of existing frameworks, this paper presents an energy based approach for estimating the time to liquefaction. The proposed method uniquely uses the cumulative absolute change in strain energy, since it is directly related to kinetic energy and has been shown to be uniquely related to liquefaction triggering. The paper also presents a novel equation for the calculation of the energy applied to a point by a travelling wave, a procedure for the calculation of the cumulative absolute change in strain energy from a broadband seismic motion and an estimation of the build up excess pore pressure and triggering of liquefaction with respect to time, throughout the depth of a soil deposit based on the upward propagating shear wave.

Section snippets

Definition

The normalised cumulative absolute (change in) strain energy (NCASE) (or CASE when not normalised), which is calculated as the cumulative change in absolute peak strain energy divided by the at rest initial vertical effective stress (Equation (3)), is graphically represented as the sum of the absolute change in strain energy between the strain energy peaks in the response (Fig. 1). The peak strain energy points (j) (local maxima and minima) can be determined graphically, or as the intercepts of

Estimation of accumulated strain energy at depth of interest

In the previous section the triggering of liquefaction was quantified using NCASEliq. In this section an exact solution to estimate the NCASE demand (NCASE*) at any depth from a free surface will be developed for an upward propagating broadband seismic shear motion.

The cumulative absolute kinetic energy (CAKE), can be computed using Equation (5) as the total kinetic energy per unit volume given and taken from a point. Equation (5) is simply the sum of the cumulative absolute change in kinetic

Numerical validation study

A series of linear, equivalent linear, nonlinear total stress and nonlinear effective stress one dimensional analyses were used to validate the NSES method for the estimation of NCASE* and ultimately estimate the build up of excess pore pressure and triggering of liquefaction.

The NSES method used the upward motion at the base, the elastic soil properties and 3% damping, and Equation (12) for the impedance. Five hundred synthetic soil profiles were randomly generated using the ranges and

Conclusion

This paper presents a novel approach to estimate the build-up of excess pore water pressure and the time of liquefaction using kinetic energy and strain energy. The main findings of this work are:

  • 1.

    The cumulative absolute change in strain energy required to liquefy the soil (NCASEliq) was shown, through the investigation of existing element test data, to be nearly independent of the amplitude of loading and could be predicted from the dissipated energy required to liquefy the soil.

  • 2.

    A unique

Acknowledgements

This paper was produced as part of the LIQUEFACT project (“Assessment and mitigation of liquefaction potential across Europe: a holistic approach to protect structures/infrastructures for improved resilience to earthquake-induced liquefaction disasters”) has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No GAP-700748. This work was financially supported by: UID/ECI/04708/2019- CONSTRUCT - Instituto de I&D em Estruturas e

Glossary

List of acronyms

CRR
Cyclic resistance ratio
CSR
Cyclic stress ratio
CASE
Cumulative absolute change in strain energy
NCASE
Cumulative absolute strain energy normalised by the at rest vertical effective stress
NCASEliq
NCASE required to cause liquefaction
NCASE*
NCASE generated by a loading (e.g. ground motion)
NCASETS*
NCASE* from total stress nonlinear analysis
NCASElinear*
NCASE* from linear analysis
NCASEEqlin*
NCASE* from equivalent linear analysis
NCASENSES*
NCASE* from NSES method
CAKE
Cumulative absolute change in kinetic

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